I'll no longer link individual fanfics as I cannot dedicate the time needed to read and list each one, but I will post sites where you can read and submit your own fanfics. Please be aware of the individual fanfic ratings.
You can narrow down the large section by characters, genre, ratings, and language.
Organized by genre, and even includes an 'Avatar dictionary' and character profiles for easy reference.
Updated fairly often and fics range from K to M. Uses tags to organize characters, pairings, ratings, and authors.
Avatar Fanfic Contest
An active weekly fanfic challenge community, with a new theme every week.
Sites dedicated to specific Fanfic series:
The Chronicles of Leo
50 years in the Avatar
future, with themes of Brotherhood, Redemption, Penance, and History repeating itself.