Introduction:
I love reading and ever since I was a little kid I've never been without a book or two. My love of reading, combined with my somewhat overactive imagination led me to writing. I've been writing stories and novels (mostly as a hobby) since I was eleven or twelve years old. Unsurprisingly, my earliest attempts weren't all that good. Some had good ideas but poor writing while others were just lousy in general. But over the years, I've improved significantly.
There are many things that helped improve my writing. First off, I got advice from others and, while I didn't blindly accept everything they said, I didn't dismiss it either and gave each suggestion careful consideration. I also took a look at the writing styles in my favorite novels and compared it to my own, trying to determine what what those authors did differently (and in most cases better) than me. Finally, I wrote a lot. Like with most things, writing improves with practice. Though writing isn't enough by itself. I didn't just write, I read my writing, thought about what could be done better, and rewrote it. While I'm not one of those overly critical people who's never happy with their work, I'm not one of those guys who thinks that everything he does is absolutely perfect and brilliant either. I've always been very good at taking a critical look at my own work and identifying its strengths and weaknesses.
Unfortunately, not everyone has this talent and, even for those who do, learning from your own mistakes takes a lot of time. In order to help out aspiring writers, and those of you who just want to improve your school projects and fan fics, I'm going to be talking about some of the common mistakes made by amateur writers and how to avoid them.

Mistake #1: Failing to Take Constructive Criticism
This is something I see a lot these days, especially on the internet. And not just in regards to writing, it happens with art and other things as well. Simply put, a lot of people just can't take constructive criticism. The moment anyone says anything negative about their work, they get angry and defensive and rant about how their work is perfect (or close enough) and no one has any right to criticise it.
This is a really bad attitude to take not just towards writing but towards life in general and will serve you poorly in school and the work place. Of course, that doesn't mean everyone who criticizes your work is correct. Comments like, "OMG this sux" should naturally be ignored and even many respectful and thoughtful critiques should be taken with a grain of salt. But if someone takes the time to give you a well thought out comment or critique you should pay attention. For example, if someone told me something like, "You keep forgetting to use quotation marks when your characters are talking," or "You need to do a better job of describing what your characters are doing in this section" I'd tell them thanks and check to see if they were right or not. If they were, I'd make any necessary changes. Furthermore, I'd be glad that person took the time to read my writing and tell me how to improve it.
Keep in mind, most people who offer critiques of your work aren't trying to be mean or pick on you, they're just offering their thoughts, opinions, and sometimes advice. That's no reason to snap at them or get angry and defensive. Some of them are probably wrong but many are probably right as well. And if you can't learn to take constructive criticism and see when your work needs to be changed, your writing will never improve.

Mistake #2: Proper Spelling and Grammar...Or Lack There Of
Despite the pervading attitude in web forums and chat rooms, proper spelling and grammar are very important. While just about everyone will forgive the occasional typo, a large number of mistakes makes your work look sloppy and unprofessional. In extreme cases, it can even make your writing nearly impossible to read. I've received a few e-mails before where the grammar and spelling was so bad that I really had no idea what the sender was trying to say.
While you may think that it's no big deal if people don't like your spelling and grammar, you'd be wrong. True, if your spelling and grammar is bad online the worst you'll do it annoy the people reading it. But if you get into the habit of ignoring proper spelling and grammar when you're posting in forms, sending e-mails, IMing, chatting in games, and the like, you'll have a harder time using proper spelling and grammar when you need to. And if your spelling and grammar is bad on something more important, like a school paper, job application, or any other professional bit of writing, you're going to find that even a few mistakes can cause major problems.
What's more, there's no good excuse for not using proper spelling and grammar in your writing (unless English isn't your native language, in which case you get a bit more slack). You learn this stuff all throughout school. Plus, pretty much everyone with a computer has a program with a spell checker (Microsoft Word, for example). Of course, spell checkers can't catch some things (like if you mix up "where" and "wear") but they'll find the vast majority of spelling mistakes and typos. If you have problems with spelling things correctly from memory, there's no good reason not to use a spell checker. Even if you're writing something for a web forum or e-mail program that lacks a build in spell checker, you can always type it up in Word (or whatever word processor you use) then copy and paste it into where ever you want it. It'll only take a few seconds extra and can seriously improve your writing.
Many programs also have a grammar checker you can use, but personally I'd avoid them, or at least take their suggestions with a grain of salt, as they tend to be wrong as often as not. Instead, pay attention to your grammar lessons in school and look at how things are written in commercial books and magazines (as they go through professional editors before being published). With a little work, you can pick up decent grammar quickly enough if you try. For that matter, I've noticed that most people speak with better grammar than they write. If you have problems with grammar in your writing, try reading what you write out loud and then think if you'd really say it that way if you were talking instead of writing.

Mistake #3: Making Things too Casual
This should be pretty obvious, but you shouldn't use chat speak, text speak, or emoticons (smiles :-) and the like) outside of texting and IMing. For that matter, your really shouldn't overuse them in texting or chatting either, as it can lead to bad writing habits. Nothing makes your writing look dumb and amateurish like a bunch of LOLs, OMGs, and :-P  l33t speak is even worse.
You should NEVER EVER use that type of stuff in formal writing. When writing a story, the only time you should use it is if your characters are using it. For example, if I was writing a story about some thirteen year old kids living in the modern world, and there was a part of the story where they were IMing each other, they'd likely be using chat speak in their IMs, so it would be ok for me to use it there, since I'd just be reporting what they said. However, if they were talking to each other in person, or if the story wasn't about people who would normally be using chat speak (for example anything not set in modern day Earth), it should be avoided at all costs.

Mistake #4: Unnatural Dialogue and Narration, The Importance of Flow
To better illustrate what I'm talking about, look at the following bit of sample dialogue between a couple teenagers.

"Hello, Dan. Do you want to hang out after school?" Fred asked.
"Sure. We could get pizza and then go watch a movie." Dan replied.
"Yes. That will be lots of fun." Fred agreed.

Now, there's nothing particularly wrong with that exchange, but take a look at this rewritten version.

"Hey Dan, want to hang out after school?" Fred asked.
"Yeah! How about pizza and a movie?" Dan replied.
"Sure, that'll be great!" Fred agreed.

Both versions are grammatically correct. But if you think about it, the first version sounds overly formal and a bit stilted. Most people these days don't talk like that, especially a couple of teenagers. It's a common mistake for novice writers, whether they realize it or not, to make their characters talk in an overly formal and stilted way. In real life, people combine a lot of sentences (using commas, "and", "but", and the occasional semi-colon), use a lot of contractions (that will = that'll, let us = let's, etc) and talk with each other in a fairly casual and not overly polite way (at least in the US, in some countries (such as Japan) people do speak in a very polite and formal fashion in many situations). Their words and sentences flow together in a smooth and natural way, at least most of the time.
While it's technically not "dialogue", narration is still essentially someone talking (even if that someone is a nonexistent third party who is describing the events) and therefor follows a similar rule. While narration is generally a bit more formal than a conversation between friends, it's still fairly informal and should have a smooth flow as well. Just look at this guide, for example. I'm using plenty of contractions and compound sentences because I'm "talking" to you, my readers, just like a narrator in a story. For another example...

The night sky was dark and black. There was not a star to be seen.

Grammatically it works, but it's kinda stilted and rather dull to read.

It was a dark night, the inky blackness completely blotting out the stars.

Doesn't that sound a lot better? Considering that most people talk in a fairly well flowing fashion, you'd assume that they'd naturally write like that, though it usually doesn't work that way. Making your writing sound natural and flow well is something that often takes a lot of practice. A good way to work on it is to read your writing back to yourself (ideally out loud) and think about if that's the way you or your friends would say it if you were talking to someone in real life. Of course, you may not want your characters and/or narrator speaking the exact same way you do, but this should still help you notice if things don't sound right.

Mistake #5: Dialogue is Everything...Or is it?
Ddialogue is an important part of nearly every story. Not that you can't write a good story without dialogue, but that's a fairly uncommon approach, especially outside of children's stories. But anyway, I probably don't need to convince anyone about the importance of dialogue. What's often overlooked by novice writers is the importance of everything else. For example, take a look at the following bit of dialogue.

"It sure is peaceful here, isn't it?" Sara commented.
"Yeah, it is." Brain replied.
"Is something wrong?" she asked.
"No, not really." he answered.

So what's wrong with that bit of dialogue? Actually, the dialogue itself is fine, it's the rest of that sample that's the problem. Or, more specifically, the fact that there isn't really a "rest of the sample". Now take a look at this version of the same conversation.

"It sure is peaceful here, isn't it?" Sara sighed happily as she dropped down on the soft green grass. A cool breeze passed over the hill, blowing a loose strand of strawberry blonde hair across her eyes as she lay there, enjoying the rare moment of peace.
"Yeah, it is." Brian's tone was cold and dismissive, a stark contrast to the warmth of the day. Unlike Sara, he looked anything but relaxed as he stared down at the city below.
Startled by his response, she looked up, a worried expression on her normally carefree face. "Is something wrong?" she asked, concern filling her voice.
For a long moment, Brian was silent, unsure how he should respond. At last he turned away, not wanting her to see the hurt in his eyes. "No, not really." he clenched his fists, angry at himself for lying to her. But really, what choice did he have? She wouldn't understand and, even if she did, there was nothing she could do change things. There was nothing either of them could do.

So, what do you think? It's like an entirely different conversation, isn't it? All the dialogue itself tells you is that Sara and Brian are talking in a presumably peaceful place. Brian might be upset, or Sara might have just misread the situation. But when you add in all those descriptive sentences from the second example, you not only find out a bit more about where they are (on a grassy hill overlooking a city on a warm yet slightly breezy day) and a little about what they look like (Sara has strawberry blonde hair), you're also able to notice all the subtext hidden in the conversation. In the second example it's obvious that Brian is troubled by something, and a bit angry as well. The two seem to know each other well (something you also can't be sure about by just reading the first example) but Brian is keeping something important from her, quite likely to keep her from worrying.
As you can see, dialogue is important, but it often doesn't come close to telling the entire story. And, even when there isn't some deep subtext to a conversation, it's still important to use some descriptive sentences (though maybe not as many) so the reader can get a feel for where the characters are, what they and/or the surrounding areas look like, and what they're doing. It also helps break up long chunks of dialogue, which can start to drag after a while. They say a picture is worth a thousand words, right? Always remember that, unless you're writing a comic or a movie script, you don't have a picture so you're going to need those thousand words instead.
On a related note, while it's important not to neglect descriptive sentences, don't over use them either. While this mistake is more often made by more experienced writers than amateurs (at least from what I've seen), it's still good to keep in mind. While you want enough description to set the scene and keep your readers informed about what your characters are doing, you don't need to go describing every building, rock, and passerby as well. The reader doesn't need to know exactly what the third house on the street looks like (a basic overview of what houses or buildings in the area look like is fine) unless it's a very important location in the story. Similarly, the reader doesn't need to know the complete life story of the guy who runs the coffee shop either (a couple sentences about his looks and attitude should be more than enough), unless he's an important character. In writing, as in just about everything else, finding a proper balance is important.

Mistake #6: Skipping the Small Stuff and Other Problems With Pacing
This is a common mistake both in written stories and in web comics. Unlike the previous mistakes, which are all tied into writing itself, this is more of a storytelling mistake. Anyway, if you look at any story you'll see that there are certain "key scenes" in it. A key scene it a scene that is extremely important to the main story line. If you don't quite get what I'm talking about, take a look at this simple outline.

Chapter 1: The main character finds the legendary sword of evil killing.
Chapter 2: The main character explores the city.
Chapter 3: The main character joins a team of adventurers to fight the great evil lord.
Chapter 4: The group takes a rest from their journey to relax.
Chapter 5: The group defeats the great evil lord in an epic battle.

Ok, so here we have the outline for our rather short and overly generic fantasy story. Take a look at the descriptions for each chapter. As you can see, chapters 1, 3, and 5 are clearly very important to the main plot and are therefor the key scenes of the story. They tell the story of the hero finding the sword, meeting his allies, and defeating the bad guy. Chapters 2 and 4, on the other hand, don't sound very important. Chances are, if you removed chapters 2 and 4 the main plot of the hero going after and beating the villain would remain intact.
So why bother with chapters 2 and 4 at all, if they're clearly not needed? Might as well just skip them and jump right to the important stuff, right? Actually no. It's a common mistake for novice writers to have their stories jump from key scene to key scene with few if any other scenes mixed in. But, while those scenes may not be crucial to the main plot, they can still add quite a lot to the story. Going back to our outline, while chapters 2 and 4 don't do much of anything to advance the main plot, they're a good opportunity to develope the personalities of the main character and his friends, show more of the world where the story takes place, and maybe even work in a bit of backstory.
If you just jump from key scene to key scene, it makes it much harder for readers to get to know the characters and really understand their personalities and motivations. Any character development will likely seem rather rushed and often disjointed, since you won't be able to gradually show them change and grow. Your world and setting will likely feel undeveloped as well since there will be little time to show any aspects and locations not directly tied into the main plot. Not to mention that if you just go from big scene to big scene without any breaks you run the risk of "wearing out" the reader.
If you don't believe me, just take a look at any good plot driven novel or TV show. You'll see that there are quite a lot of chapters or scenes that have little to no real importance to the main plot but instead serve to further develop the characters and setting and offer a change of pace from the more intense key scenes.
There's a partial exception to this rule when it comes to short stories. Because of their limited length, you need to keep the focus mostly on the main plot. While it's still good to work in a few less important scenes if you can, they should be rather short and few in number (perhaps only one or two) in order to maintain the appropriate pacing.
On that note though, it's important not to overdo it either. Like with the previous mistake, you need to find a balance. If your key scenes are too close together, you'll have the problems I previously mentioned. But if you place them too far apart, your story will drag and, no matter how much people like your characters and setting, they'll eventually get bored waiting for the plot to advance. No matter how good the main plot is, without proper pacing, it's going to suffer, sometimes significantly. The only real way to improve your pacing is to examine good stories, try to get a sense of their pacing, and keep it in mind when writing your own stories. In this case, studying and practice make perfect.
The main exception to this is stories which don't really have a main plot like most TV sitcoms and some "slice of life" type stories. In those, the whole point of the story is merely to follow the characters and see what they happen to be doing, whether it's big and important or not. However, since those type of stories lack a main plot, extra care needs to be paid to the jokes and/or characters to ensure that they can maintain the reader's interest for a long period of time on their own strengths alone.

Mistake #7: Unbelievable Characters: How People Should and Shouldn't React
When I say unbelievable characters, I'm not talking about someone who can fly, cast magic spells, or transform into a t-rex. The thing is, no matter what world people live in or what strange powers they have, you can count on most people to react in fairly similar and predictable ways. Naturally, there's some variation based on different peoples' personalities and lives but, a few extreme examples aside, it's still very easy to predict how someone will react in various situations as long as you know a little bit about them. In Mistake #4, I already talked about the importance of making your character talk in a natural and believable way, but all that's useless if they don't act believably as well. Take a look at the following example.

"Jim! Jim! Did you hear?" Brad burst into the workshop, sending the door crashing against the wall. "The princess was found murdered in her room this morning!"
"Really? The Princess?" Jim looked up in surprise. "I never would have guessed that someone could have snuck into the palace and done something like that. I wonder who did it. The whole place must be in an uproar now." he sighed then turned back to his work.

So, what's wrong with that bit of dialogue? Absolutely nothing, but only if we make certain assumptions about Jim. If Jim is just some average guy who lives in the kingdom, his reaction is perfectly normal. After all, he wouldn't be close to the princess. He may have never even seen her. Sure some people, like Brad, will be a bit more excited and curious about the events but even so, for them it's just a bit of exciting news, not a big deal. But what if Jim and the princess are friends? Maybe they were even in love. In such a situation, Jim's reaction would be completely unbelievable. No one without serious mental problems would react like that to the death of someone close to them. If that was the case, her death isn't something Jim could just brush off with a single comment and return to work. He'd be shocked and sad at very least. If they were very close, he'd be absolutely devastated. Her death would effect him for months or years to come, maybe even his entire life. On a slightly less important note, if Brad knew of Jim and the Princess's relationship, he probably wouldn't just burst in and blurted out the news either, he'd be much more depressed and sensitive about it. Let's look at another example.

Light swirled through the air as the stranger raised his hand and sent a shimmering red ray streaking towards the monster on the other side of the clearing.
"Hey, he's using magic." Jackson murmured as he watched the battle unfold from his hiding place behind a large rock.

Well, can you spot the problem? This one should be easier. To put it simply, unless he frequently runs into magicians and monsters, Jackson's reaction is completely unbelievable. Even if Jackson lives in a fantasy world where magic and monsters are fairly common, he should probably be somewhat surprised, scared, and/or excited (depending on his personality, profession, and just how common that type of stuff is in his world). And what if Jackson is an ordinary guy from the modern day US who ran into the magician and monster while hiking in the woods? In that case, the above reaction would be a million times worse. If a person sees something strange or impossible, they don't just casually accept it. They might get really scared or really surprised and they'll quite likely try to convince themselves that it can't possible be real. The one thing they definitely won't do is just go "ok" and be fine with it.
Both of the above examples are a little extreme and many of you are probably thinking that this is pretty obvious, but I've seen lots of novice writers make this exact mistake in very similar scenarios to the ones above. And if it's that easy to mess up in such extreme situations, just imagine how easy it is to make this mistake at other, less blatant, points in a story. When writing how your characters react to different situations, try to imagine what you would say and do were you in their shoes. Naturally, your characters may have different mannerisms and personalities than you, but your own feelings can still make for a good starting point. If characters act in extreme ways to something that shouldn't matter to them or just calmly accept life changing events, they lose quite a lot of their believability and readers will have a much harder time forming a connection with them. Subconsciously, most of us have a good idea of how people would or wouldn't act in different situations. As such, when we see characters acting strangely, our minds will pick up on it and reduce our immersion in the story even if we can't consciously point out what's wrong.
Writing realistic and believable characters is something that comes naturally after a while but many novice writers (and even some long time writers) need to spend a lot of time stopping and thinking about. As the characters are, in most cases, the most important part of a story, if they aren't believable the entire story will suffer greatly.

Mistake #8: Everywhere isn't Just Like Home
The best way to explain what I'm talking about here is, you guessed it, with a bit of sample dialogue.

"Tom, stop beating around the bush! Do you want to go or don't you?" James groaned in exasperation.
"To Olive Garden? Of course I do. But it's like, five miles from here, my parents won't want to give me a ride, and I only have around eight dollars anyway."

Well, that seems simple enough, right? No problems as all...assuming that it's taking place in the modern day US. But what if this was supposed to take place in Japan? Obviously, James and Tom aren't Japanese names, the Japanese currency is the Yen, not the Dollar, and (as far as I know) Japan doesn't have any Olive Garden restaurants. A bit less obviously, Japan measures distance in kilometers, not miles, and "beating around the bush" is not a Japanese idiom so it would be very unlikely for a Japanese person to use or understand it. Finally, in many parts of Japan most people get around using bicycles and/or public transportation so Tom's comment about getting a ride from his parents might not fit either. And that's just from two lines of dialogue. Even though you may be writing a story in English, people in other countries talk differently, live differently, and have different idioms and manners of speech. They'll also react differently in certain situations. Everything isn't just like it is in your home country (or even your home town) so you've got to take that into account, research the place where your story takes place, and make sure that your characters, terms, and the like are appropriate for the situation. And even then, that's just modern day Earth. If you mix in other time periods, things change even more. The US of the 50's, for example, was a very different place than the US of today. Technology, stores, lifestyles, mannerisms, and the like were all considerably different back then and you need to take that into account if you're writing a story that takes place in that time period.
You also need to be careful if your story takes place on another world. Other worlds are not Earth and would naturally have their own currencies and units of measurement, so you should probably avoid, inches, miles, kilometers, and the like. They'll also have different names for the days and months (and may even have different numbers of days and month). And, even if the people on another world look and act like Earth humans and talk in a fairly similar fashion, they're still going to have different terms for some things and different idioms as well. Chances are, most people wouldn't have common "Earth names" as well, so James and Tom may be out. Of course, there's a fine line between making things different enough to clearly differentiate your fictional world from Earth, and making it so different that readers have to constantly check an appendix (assuming you write one) to find out what all your new terms and words mean. Like with everything else, it takes thought and practice to figure out exactly what should and shouldn't be changed. It also varies a bit by story and target audience.

Conclusion:
While this is hardly a comprehensive list of potential writing mistakes, it should at least tip you off to many of the more common ones made by novice writers. But, in the end, the best things you can do to improve your writing are to read lots of well written books in all styles and genres and then practice, practice, practice. And don't just write something and move on. Instead, reread and edit your writing while keeping and open and somewhat critical mind towards what could be improved.
Like any skill worth having, the ability to write well takes a lot of time and effort to develop. Not everyone can become a great storyteller. To a certain extent, the ability to create interesting characters and stories is something that you either have or you don't. However, with enough effort, just about anyone become a good writer. And that's a skill which will serve you well in many ways all throughout your life.