Super basic explanation is this: stories written by fans of a particular book, movie, game, series etc. They usually include existing characters and settings. Usually found on (but not limited to) the internet.
So, now that you know what it is, let me dispel some myths about Fanfiction for us all.
1. Fanfiction is always romantic.
I cannot even begin to tell you how wrong this is. Fanfiction is awesome because it can be any genre you want! You're not limited to writing slash fiction (a focus on developing a relationship between characters).
2. Fanfiction is only written by teenage girls.
Nope. I know plenty of folks who write 'fanfics' who aren't still in high school. If there's a character or narrative out there that you love and you want to write about, then do it. I started writing when I first started high school but I still write to this day (I'm 24).
3. It's limited to one genre i.e. Vampire Romance.
I guess that myth goes hand-in-hand with the two before it but again, it's just a big myth. There are heaps of different genres out there for people to write about, you're only limited by your imagination.
4. It's a time wasting hobby that gets you nowhere.
Not true at all. Just look at Fifty Shades of Grey…
Okay, not the best example, but a creative hobby is never a bad thing and continued practice will help you improve your skills. Who knows, you may end up writing a novel some day of your own.
5. Fanfiction is poorly written.
Let me tell you that some of the best pieces of writing I've ever read have been Fanfiction. Just because it's not printed on paper or being sold in a bookstore does not mean that it's badly written. Sure, finding a good author that suits your taste can be challenging considering the number of fanfics out there, but if you persevere you will find something you like.