There are some mistakes that I see with incredible frequency.  Most of them have to do with characterization and plot.  I'm not a strong fiction writer, so I'm not comfortable in speaking to those mistakes.  I'd suggest seeking the wisdom of experienced authors for that.  However, there are a lot of common mechanical errors out there.  Here is the list of the top 9 mechanical mistakes I see in fanfiction.  Why the top 9?  Because someone has to balance out the Nostalgia Critic.

Mistake 1: "Per say".  "Per se" means "in itself", "per say" means nothing.  "Persay" is also wrong.  Please, open up your stories in an editor with find-and-replace capabilities, and replace all instances of "per say" and "persay" with "per se".  Using the wrong word here does not make you stupid per se, but it makes you look stupid.

Mistake 2: "Might".  This shows up a lot in Applejack's speech.  "Might" means many things.  It could mean "possibly", or "power", or "great strength".  What it does not mean is "a little".  If you have Applejack say "that might be a might difficult", you are wrong.  The word you want is "mite".  It might be a mite difficult to remember this, but you look like a fool if you use the word "might" improperly.

Mistake 3: Apostrophes.  ' <--this is an apostrophe.  It does not mean "Look out, here comes an 'S'!" If you use an apostrophe for pluralization* , or whenever a word ends in an s, your writing is bad and you should feel bad.

Mistake 4: your/you're.  This one is very easy, so you should never make this mistake.  If you aren't 100% certain about whether to use "your" or "you're", take it out and put "you are" in its place.  If that replacement works, you want "you're"", otherwise you want "your".  This test is guaranteed to be accurate, or your money back!

Mistake 5: there/their/they're.  If you're not sure, try replacing it with "they are".  Does it work?  Great, you want "they're".  If it doesn't, you need to figure out whether you want to use "their" or "there".  Are you talking about a location?  Use "there".  Do you mean "belonging to them"?  Use "their".   It's really quite easy!  Twilight and her study buddy left some books in the library, and now they can't read them!  How do they feel about it?  "They're upset that their books are over there." Simple!

Mistake 6: then/than.  This one is simple.  If it's a comparison, use "than": "Twilight Sparkle reads a lot more than Rainbow Dash does".  If it's a sequence of events, use "then": "Twilight went to Sugarcube Corner for lunch, then she went home and reorganized her books".

Mistake 7: its/it's.  "Its" means "belonging to it".  "It's" means "it is" or "it has".  Remember this.  I haven't come up with anything easier than straight memorization here, sorry.

Mistake 8: Dialogue tags.  "Said" is perfectly okay.  It's practically invisible.  "Asked" is much the same.  If you want to mix it up, mention your characters actions instead of using "said" or "asked"--any action attached to a name in the middle of a dialogue is enough to tell the reader who is speaking, you don't need to explicitly say it with a verb.  Generally, though, your characters should not habitually explicate, demonstrate, agree, murmur, inform, question, query, say bluntly, reply grimly, ejaculate, protest, declare, demand, or anything of the sort.  Show (with dialogue and body language), don't tell.  Those telling sorts of dialogue tags are okay in moderation, but like strong spices, they should be applied with a light hand.

Mistake 9:  Tense confusion.  For the love of Celestia, write your story entirely in the same tense.  Don't switch tenses between paragraphs unless you know exactly what you're doing.  Don't switch tenses within paragraphs, ever.  If you're not sure what to do, just write everything in the past tense, except for dialogue (the tense of dialogue is independent from the tense of everything else).  You can write in the present tense, but I find it easier to write in the past tense.  When everything is in present tense, it's hard to discuss sequences of events in a way that feels natural.  If you want to write in the present tense, go ahead, but be careful not to slip into the past tense inadvertently.

*It is acceptable to use an apostrophe to pluralize a single letter: "dot your i's and cross your t's".  However, it is even better to quote the letter--dot your 'i's and cross your 't's--, or to change the font style--dot your is and cross your ts--as this is consistent with other ways of specifying that you are mentioning, as opposed to using, a word.  This is largely irrelevant, as in most fanfic you won't be on the mention side of the use-mention divide very often, but I figured I'd throw it out there anyway.