I like writing fanfic because it gives me a chance to hone my writing skills and I love certain characters and want to see more happen with them. I love reading fanfic because it gives me a chance to spend more time with my favorite characters and see what other writers do with them.
I've started a handful of Labyrinth stories (and actually finished one). I started one Harry Potter fic that I may or may not finish. I'm currently doing research for a different one, and I have story ideas to spare. What can I say, plot bunnies like me.
Speaking of plot bunnies, I'm putting together a list of frequently used fanfic terms. I'll probably post it tomorrow.
The best place to find fanfiction, IMHO, is Fanfiction.net, probably the largest general fanfiction archive on the Internet. The quality of the stories there varies from "Who let this person pass English?" to "Why isn't this person a professional writer?" Most of them are somewhere in between. Mine certainly are. FF.net (as I tend to call it) has categories for just about every book, movie, TV show, etc. you can think of. As far as I know, the largest category there is for Harry Potter fanfiction, with over 75,000 entries.
If you're looking for other fanfic sites, a great place to search is Webring.
Not all authors approve of fanfic. Anne Rice is probably the most famous of the ones that don't. She forbade all fanfic based on her books, though she does allow fanart, which is artwork done by fans that is based on their favorite books, movies, etc. Why she allows one and not the other is beyond me. JK Rowling is okay with Harry Potter fanfic as long is is neither offensive nor commercial. I once heard that she's amazed that fans are interested enough in her characters to write their own stories about them.