I read fan fiction strictly in the Twilight fandom. I'm not an author, just an overly-obessive reader. You may seek out my recommendations in My Favorites. They're purely must-reads; you'll enjoy them.
Here are some good stories from Twilighted, as well:
The Dreams of The Boy In The Nighttime by Char
Summary: Bella Swan has had a tragic and abusive childhood; everyone knew it. But not everyone knew of the bronze-haired boy who visited her in her dreams. He was the only one who knew the true Bella. But Edward wasn't real, was he?
Leaf Dances by Shiloh
Summary: When Edward Cullen first moves to Forks, his neighbor Tyler Crowley makes it instantly clear that he demands privacy for him and his insane wife, Isabella. But the longer Edward is witness to Bella's bizarre behavior, the more questions he has, and the less willing he is to remain uninvolved. AH-AU, OOC, BxE, canon pairings, lemons, angst.
Green, Red, Gold by Segolily
Summary: Edward Anthony Masen Cullen. 1918-2005. Take a journey through his life.
Will You Be Ready When It Comes? by Hopeful Wager
Summary: Are we victims of a fate, or makers of our destiny? Can one choose to deny a nature so inherent as bloodlust? Is there a choice to be made between life and love, and hope and despair? Live, Die, Drink, Eat, Laugh, Cry, Hope, Fight, Hunt, Try, Grow, Sigh... but mostly, live. A little drama, a little romance, a twist of lemon and lime, and the occasional smile. Post BD.
(By the way, some of those may actually be on FF.net, but I'm not entirely sure.)