What are they:  Kazaa is a peer-to-peer file exchange programs.

What they do:  The are used to exchange files of both copyrighted and public material.  A large majority of the material exchanged is copyrighted.  There is no central discovery server.

Are They Popular:   We did a study in the P2P group at Stanford two months ago... with pretty interesting results. Kazaa consistently topped out at over 50 terabytes of data and peeked at just over 1.1 million active clients.

Kazaa is Legal:

First, the courts in the Netherlands rule against Kazaa, in a higher appeal, the courts rule in favour of Kazaa saying "The software itself doesn't commit any illegal acts, it's the users that could do that, therefore the software is legal"  The Judge ruled that even if the users violated copyright, Kazaa did not. Important in the ruling were the facts that Kazaa does not rely on a central server, and that it is not bound to music or video files alone.