There many different types of Twilight fan fiction.
Canon - Stories that are set before, during, or after the Twilight books. The tales fit in with the plot of the Saga and the characters act in conformance to how Stephenie views them.
Human - Some or all of the vampires and wolves are human. Or Bella is a vampire and the others are human. For example, Bella and Edward are childhood friends; Edward is the new kid at school and meets Bella, Miss Popular. These typically try to keep the basic personality quirks of the characters but imagine how they'd be different as humans.
Alternate-Universe - Partial or complete departures from the Twilight canon. What if Bella falls for Jacob...or Alice? What if Edward is a police officer and saves Bella from an ongoing crime? What if Edward is a wounded Civil War soldier and Bella is his nurse?
Crossovers - These pair Twilight characters with characters from other stories. Harry Potter is the most popular cross-overs, but I've also seen stories featuring Vampire Diaries, Vampire Academy, Mortal Instruments, and more. I'm really curious about the more unusual pairings, such as Fruits Basket (my favorite manga!), Baby-sitter's Club, and Gone With The Wind.
These are very important to me. A lot of Twilight fanfic is extremely graphic. Either Mature or NC-17. I prefer Stephenie's version of "graphic sex," so I am careful to not read any M or NC-17 fanfic. I read G through PG-13 stories.
Some Fanfic I Recommend
I go in spurts with fanfic. I haven't read much in a long time, but when I start reading them, I read them actively for weeks. I only read Canon fanfic tales. Perhaps it signifies a lack of imagination, but I don't want to read a story unless it could have conceivably occurred in the Twilight Saga. Also, with a few exceptions, I am picky about writing quality. If I think the writing is bad, I'll give it up after a few sentences - unless it's really funny. And I rarely read fanfic more than a few chapters. I just don't have the time to devote to long fanfic when I could be reading a book. So all my snobbish prerequisites aside, here are some stories I recommend: