Fan fiction, or fanfic, for short, refers to stories (and sometimes poetry), that use the characters or settings of an existing, copyrighted work. In other words, they're stories set in the same world as a TV series, movie, or book. They may be written about the characters everyone knows from that world, or about completely new characters, just using that setting.
Fan fiction isn't exactly legal, since the people who created the TV show/movie/book own the rights to the characters and settings. Fanfiction writers are usually breaking copyright laws, though there are some fanfics that might be legal, under certain circumstances. However, many authors and producers either turn a blind eye to fanfiction, or don't mind it, as long as no money is being made by the writers, and they're just sharing their stories with friends.
Relationship fiction is a very popular type (and mostly what I write). Many fans like to read love stories, romantic comedies, romantic adventure, about characters they enjoy in tv shows and books. Not all of the stories about couples are happy Harlequin romances, of course. Some, while they might be about love, or sex, aren't all that romantic at all.
Unconventional Relationships: Some of the most well-liked stories seem to be those containing pairings that don't happen in the original show/book/movie. In Buffy the Vampire Slayer fan fiction, for example, there are plenty of Buffy and Angel, or Buffy and Spike, love stories, but there are also many romantic stories about Buffy and Xander, who've never been a couple on the TV series. But wait, there's more. (You knew I was going to say that, right?)
Slash describes a specific type of romantic/sexy fan fiction -- stories in which the couple is male/male, or female/female. Those are usually considered "unconventional relationship" stories because most tv shows don't have gay couples on them -- though there are some, like Buffy's Willow and Tara.
The word "slash" originally comes from the shorthand that fan fiction writers used to describe who the members of the romantic pair are: "name/name". On the surface, this doesn't make a lot of sense, if you've read much fanfiction, because most writers use that "/" symbol for boy/girl pairings, too, right? Like "Buffy/Angel." "Lana/Clark." Watch me shrug. Yes, it doesn't make much sense, you're right. The legends say, the first use of the "/" was to describe male/male couples, so the word "slash" became associated with male/male, and later with female/female, as well. Not having been around back then, I can only take people's word for it. Simple answer: when someone spells out the word "slash," you know it's a same-gender couple.
Slash vs. Smut: Regardless of what some people assume, not all slash stories are NC-17 rated; some are pretty tame. I have one story on the Pointy Sticks page that I'd consider G or PG rated; there's nothing in it but a kiss, and some handholding. Nothing you couldn't see in a Disney movie -- except it's between two teenage boys. I also have stories on my Buffy and Harry Potter pages that are definitely NC-17, and not for the eyes of anybody who isn't supposed to be reading smutty things. Because they are smutty things.
SO...
Not everybody agrees that handholding and a kiss is something their kids should be reading, however, if it's between two boys or two girls. If you're under 18, you're pretty much stuck with what your parents think is okay for you to read. If they don't want you reading anything slashy, then, don't. Or at least, don't tell me about it. And if it says "NC-17," it contains stuff that you couldn't watch on TV, and your folks probably wouldn't want you watching in a theatre. So when you see "Keep out, no kids allowed," there's a reason. If your parents don't mind you reading it, that's between you and them.
If you're over 18 and you just wanted to know what the heck I was talking about on the previous page, that was it. The whole shebang. Now you know what the warnings mean. Read it or don't. If your child's been reading it, and you don't want him to, tell him so. The warnings are here. The explanation is here. It's not my job to police the internet for you so that you don't have to pay attention to what your kids are doing.
If you're over 18 and you know what slash is, and you just clicked your way over here to snicker at my ham-fisted attempts to define it, snicker away. Just keep in mind that (somewhat against my better judgement, and subject to change) I'm linking to all my stories from my main website, which means I could have family members ambling over to Pointy Sticks, including some minors who need to know what they shouldn't be reading, and why. Hence my attempt to make the definition more-or-less kid-safe. Cheers.