Bittorrent works by chopping up the data into tiny pieces. Unlike regular file transfers the data is not sent in an ordered sequence - that is sending the first piece of the file, then the second, then the third, then... etc. Instead the chunks are sent out in whatever order is in demand.
Like other P2P (peer-to-peer) file sharing systems no data is stored on a central server. Instead only "contact information", storing the IP of the "content sharers" is uploaded to a server called a "bit-torrent tracker".
Afterwards this data can be downloaded by a user in the form of a bit-torrent file. When a client - in BT called a leecher - starts to download data he's given a random piece. When the piece is downloaded he also - immediately - becomes a new source of that piece.
This achieves something remarkable. Normally in server-client system, the more clients you have the slower the download will go. Normal P2P systems could mitigate it by "balancing the load" between all available sources of the file.
Bittorrent goes even further - the very moment you start downloading you also become a source. So the more people use a torrent the faster it will eventually become.
In this animation, the coloured bars beneath all of the 7 clients in the upper region above represent individual pieces of the file. After the initial pieces transfer from the seed (large system at the bottom), the pieces are individually transferred from client to client. The original seeder only needs to send out one copy of the file for all the clients to receive a copy.
Wikipedia entry:
BitTorrent (protocol)