I have a constantly running server that would be a perfect on which to run torrent downloads. The only catch is that the server is headless -- it has no monitor or keyboard. To overcome this obstacle, I use rTorrent as my torrent client, and GNU Screen so that I can disconnect my terminal session and leave rTorrent running.
BitTorrent is popular peer-to-peer sharing communication protocol used for transferring all kind of files over the Internet. Two of the most popular BitTorrent clients for Linux are Azureus and KTorrent. If you're looking for a robust, fast, simple, and powerful BitTorrent client, you will probably go with KTorrent. If you want a Java-based client that runs on every platform and allows you to configure every detail for BitTorrent transfer, consider Azureus.