Creepypasta is a genre of internet fiction; it might best be described as short horror pieces, although they are not always short, they are often more creepy than scary, and 'pieces' must be taken in the widest sense possible, as creepypasta can include drawings, .gifs, films, and what can only be described as pranks.
In the most general sense, creepypasta is a internet page or file that is designed to unnerve or spook the viewer. Traditionally it refers most commonly to stories and pictures that float anonymously around the internet, and the 'pasta' refers to copypasta -- AKA cut and paste. However, there are many sites dedicated to producing new creepypasta, and it is not at all uncommon for creepypasta to be credited and copyrighted. While defining creepypasta is difficult, giving examples is easy enough:
The Slender Man: A series of photoshopped images and videos of a tall, slender man, who is literally faceless. He stalks his victims, sometimes for years, before luring them into the woods and killing them. He is perhaps the most famous of the creepypasta creatures, although he has been around only since 2009.
YouTube, further reading.
SCP-173: A strange creature that can only move -- and kill -- when no one is looking at it. A prime example of good, short, creepypasta, it consists of a very odd picture accompanied by a short text. It orriginally appeared on 4chan's /x/ board, and was so compelling that it gave birth to the SCP Foundation, an extensive collection of creepypasta and related works.
SCP-173 and The SCP Foundation.
Nightmare Fuel: A hazy concept, but Nightmare Fuel most often refers to a picture or video that might give you nightmares -- simple enough. Purists may insist that to be nightmare fuel, the effect must be unintentional.
Accidental Nightmare Fuel.
Screamers: Low on the totem pole, but quite familiar, screamers are any video or flash animation that starts out appearing innocuous, and ends with a surprise... generally, a monster or bloody face popping up and screaming loudly in the last few seconds. In order to enhance the effect, the viewer is usually told to turn up the volume, or more sneakily, the audio is initially very quiet but suggests that it is worth hearing, causing you to turn up the volume without being told. A cheep trick indeed.
No, I'm not going to link to a screamer.
Smile.jpg: An example of what is commonly know as crappypasta -- creepypasta that doesn't quite come off right. A photoshopped picture of a husky dog in a dark room... smiling with human teeth! The accompanying story is somewhat creepy, but it is generally held that this is not really creepy enough to justify its frequent reposting. Which is interesting, because if it isn't creepy, why does it keep getting reposted?
The smile, the story, the critique.
Spengbab: and if you thought that was bad... Spengbab is an example of parodypasta, using creepypasta to mock either creepypasta, or more often, something you hate. In this case, Spongebob. Spengbab is, in fact, not one tale or video or picture, but an entire sub-genre, all dedicated to hating and harming Spongebob. Because it's the internet, that's why.
A collection of Spengbab... stuff.
This is obviously only a very fragmentary overview of the field. There are creepypasta pages dedicated only to drawings of fantastical monsters, collections of zombie stories, or videos of ghosts. Some stories approach novellas in length, some are truly microfiction. And, of course, a lot of them are just crappy. If you have some creepypasta here on E2, please consider adding it to the creepypasta category, and of course, link it here.
Sources and Further Reading:
Encyclopedia Dramatica: Creepypasta.
Creepypasta.com. A blog with user-submitted creepypasta.
Tumbler: Creepypasta. An excellent source for an quick dose of creepy pictures and stories.
The Creepypasta Wiki.