These are some of my favorite fanfics:
You've Kept Me Waiting
You've Kept Me Waiting is a fanfic about Alice and Jasper, before they met the Cullens, when they first found each other. This was an extremely well written fanfic - the author was asked numerous times if she was SM herself.
Life's Little Choices
Life's Little Choices is based on Edward and Bella being human. The characters are a little different, but I thought the writing was good. It's a little rushed at the end, but it's somewhat long, so I can see why. The overall story is pretty good though!
The Lion and the Lamb
I'm not sure if The Lion and the Lamb should be posted under "completed" - the author said she could not finish the story due to personal problems, and it was on "permanent hiatis" (sp?) I don't know if this will ever be finished because of that note, but I'm assuming it is.
The story is practically Midnight Sun - it's everything in Twilight, up until the first kiss, from Edward's POV. It's really incredible, and it's a shame it won't be finished!
Nightfall
Wow, I just finished this. This one was written in 2006, so it was before New Moon and Eclipse came out. That being said, the author was incredible - she wrote a storyline about Bella eventually becoming a vampire, a war with the Volturi and werewolves....This one had some crazy plot twists and turns. To be honest, it's very similar to what I could see Breaking Dawn to be like. At least that's what I'm hoping for. But this was really good. and I would highly recommend it because of the similarities it has to the original Twilight series of books. Rated teen for some violence and some sexuality - but there isn't any lemons in this. The author does a wonderful job of being very sensual without saying much.