The wiki link is
http://en.wikipedia.org/wiki/BitTorrent_(protocol). It is a peer-to-peer protocol where all clients both download torrents and make then available to others (upload). This makes it very efficient in total network bandwidth, but many ISPs don't like it because of its uploading.
Torrents are not necessarily violating copyrights, but many do. However, there are also many legitimate uses of torrents. To download a torrent, all you need is a BitTorrent client.