A Bittorrent is a peer-to-peer filesharing protocol which has advantages over normal P2P like Gnutell, eDonkey and (formerly) WinMX. A server (tracker) keeps a list of all torrents organized over it, and once you download a small .torrent file it'll communicate with the tracker it was hosted on to make sure you get the file from as many people as possible at a time.
As for the clients, I'd suggest BitComet or BitSpirit. They're both user-friendly. Once you're more comfortable with torrents, you might switch to uTorrent (microTorrent).