The simple version: Tor provides a distributed network of servers("onion routers"). Users bounce their TCP streams (web traffic, FTP, SSH, etc.) around the routers. This makes it hard for recipients, observers, and even the onion routers themselves to track the source of the stream.
The complex version: Onion Routing is a connection-oriented anonymizing communication service. Users choose a source-routed path through a set of nodes, and negotiate a "virtual circuit" through the network, in which each node knows its predecessor and successor, but no others. Traffic flowing down the circuit is unwrapped by a symmetric key at each node, which reveals the downstream node.
http://freehaven.net/tor/ [main page]
http://freehaven.net/tor/slides-codecon04/ [codecon slides]