A demo is a form of computer program that achieves something remarkable (usually around computer graphics) in a very small program size.

A demoscene would be the collection of people who create these programs, and who appreciate them.

There are strong elements of informal competition -- who can shave a few bytes off this routine? -- and there are formal competitions.

The Pouet.net faq gives a nice definition "Demos are the things that happen when people who are interested in computers sit down together and make something fun and spectacular and challenging to program. They work on it until it becomes a final work of art, then they pack up their sleeping bags and travel to a demoparty where they enter it to a competition. That's pretty much the gist of it."

Pouet.net is a good source of information about demos and has a large collection.