When I hear fan fiction, I think, 'someone likes a story or characters so much they write a short story or an epilogue or something, using those characters.' For example, Gone with the Wind had an ending that left us all in the lurch, is Rhett coming back? Can Scarlet convince him that she finally figured out that she really loves him? etc. So as a fan I would write, for my own enjoyment, a short novella or story concluding the book the way I would love to see it ended, with Rhett and Scarlet together and happy. But apparently that's not it, because Fifty Shades doesn't have anything whatsoever to do with Twilight. It's actually very different. Where Twilight had no sex at all, Fifty Shades is chock full of it. So, how in the world does that equate to fan fiction?
So this is what I found on Wiki to define fanfic. "Fan fiction is defined by being both related to its subject's canonical fictional universe and simultaneously existing outside the canon of that universe." So it appears as though I was generally correct about what fanfic is, but I don't see how Fifty Shades falls into this at all. But, I have to admit I haven't read those books, so maybe there is a link that I just don't know about. Is Edward a co-worker of Christian Grey and they say hi at the water cooler every day? Who knows. If you do know feel free to tell me.
Well I hope this clears it up for all of you, and you now have a better understanding of what fan fiction is. I know I do...