Fanfictions are stories (fictions) written by people who like a series (fans - see?).
How these turn out is very varied though.
Some people see a series and want more of it, so they read or write fanfics that take the characters and setting of the series and think up new plots of the same tone as the canon ones.
Some people write little short stories (drabbles or vignettes), concentrating on moments of character interaction. Often these are romance-based, but you do get other types...occasionally.
And some people just want to write romance and porn. These tend to be the iffiest when it comes to the writer getting the characters right. Especially the slash fic (in my opinion, though many het writers are just as bad)*.
There are also AU (Alternate Universe) fics, which are very hit and miss. In one type you take the characters and put them in a different plot and setting. Very tricky to pull off because you have to explain why the characters are like they are without the circumstances of the original setting. Like explaining why Sousuke is so military-minded if he had a different childhood. However, inexperienced fan writers aften can't handle that detail, and (especially in the case of High School AUs) the characters ended up like pale shadows of their real selves.
Another kind of AU is more like an Alternate History deal - how would things have gone is something had happened differently. One interesting one I'm reading at the moment is 'What if Sousuke was a girl?' which seems to be pretty well-done.
All these types of fanfiction can be and are plgued with thigns called Mary-Sues or Gary-Stus. These can be original characters (or even existing ones sometimes) who act out the author's fantasies - like 'being sucked into [name of fandom here] world though the TV' a 'la Tron, meeting the characters, wowing them with their super-specialness and often romance ensues.
In the Full Metal Panic fandom, I find that an author on
Fanfiction.Net called dave-d writes the best FMP fanfiction. He's heavily into the Sousuke-Kaname pairing, but he writes excellent plots to go with the romance, which happens around the plot, not the other way around. The only problems are a tendancy towards Tessa-bashing, and a phenomenon I call 'Super!Sousuke'. Sosuke becomes a fountain of military knowledge, like military history, and produced lectures on whatever the subject is in many fics. The author did the same in the early Naruto fics of his I read (before I got tired of all the Naruto-Hinata he wrote). He tends to write Kaname as being in the wrong whenever there's a misunderstanding, though Sosuke is just as often the wrong one in the series.
All these problems aside, dave-d is worth a read if you fancy some fanfiction once you've run out of the actual series to read/watch.
* Slash is a romantic couple involving two men most often. In Japanese-based fandoms, it's known as shounen-ai (boy-love) or yaoi. Het is a regular man-woman coupling. Yuri is woman-woman, though that's the Japanese term. I think it's called fem-slash in non-Japanese fandoms.