What is it?

Git is a fast, efficient, distributed version control system ideal for the collaborative development of software.

GitHub is the easiest (and prettiest) way to participate in that collaboration: fork projects, send pull requests, monitor development, all with ease.
How does it work?

Get up and running in seconds by forking a project, pushing an existing repository, or starting fresh. No approval is necessary.

GitHub was written for public, open source projects and private, proprietary codes — if you use Git, GitHub is for you.
What else does it do?

GitHub provides pre-rolled post-commit hooks (IRC, Jabber, Email, Trac, Campfire, etc.) as well as an innovative web hook system for writing your own.

Every repository comes with SSH support for pushing and pulling. Private repositories enjoy full SSL support on the web side, as well.