FanFiction.net
You can narrow down the large section by characters, genre, ratings, and language.

BlueSpiritGirl Library
Organized by genre, and even includes an 'Avatar dictionary' and character profiles for easy reference.

LiveJournal.com:
Avatar Fic
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.