I'm here only for reading, but I write original stuff, mostly poems, short stories and some non-fiction, but I had an attempt at a fantasy novel a couple of years ago. Maybe I'll finish it some day, but unfortunately it doesn't look like that ATM. Though my native language is German, I write in English (I hold a degree in the language). Oh, and I guess that quite a few of the authors here could be my children.
Of course I only read fan fiction of stories I have read in the original. These are mainly Harry Potter, The Wheel of Time and Lord of the Rings.

Things I like in fan fiction:

Real AUs (see below)
Crossovers
Attempts at scenes left out in canon
Detail
Chapters with 5 - 10 K words

Things I don't like:

Self inserts
Authors who start five new stories a week and abandon all of them after two or three chapters
What if - summaries
First person narration in fan fiction, especially if canon is third person narration
Next generation copies of the original series
21st century war technology in fantasy settings
Canon books being read by the characters
Pseudo AUs where events and timeline are exactly like in canon, just with different persons or even only circumstances
Authors who whine around that they're not native speakers of the language they write in. If you're not sufficiently fluent, write in your native tongue!
Too long chapters. I can't imagine anyone but pensioners and school children in their holidays having the time to read 25 K words in one go. So split your chapters up to normal sizes please. There are certainly enough POV switches that allow this!
Slash for slash's sake. There's nothing wrong with homosexuality, but in a story it should serve a real plot purpose.
Harem. Women are human beings, not items to collect! Also, everybody wants (and normally deserves) love, how is that supposed to work if one takes all?

I would also appreciate if every author used spellcheckers and beta readers. Everybody makes the odd mistake, but having 100 of them in 2K words makes the best plot idea unreadable. That said, only those who are absolutely firm with their language should apply for beta reading!