Okay, a torrent is an example of a peer-to-peer sort of system, or p2p. That means you, the leecher in this case, need to get a torrent file. The torrent file holds information on what it is you're downloading, who is tracking the torrent (Ie: which server is the sort of operator that connects you to the person who has the file you want), and any other download info. Using a client that can download torrent files, you would open the torrent and begin to download from the person who has the files (called a "Seeder").
Since torrent files are downloaded a specific way, you can have multiple seeds sending to one leech different pieces of the file in a proper order. You can move the incomplete file to another machine and resume the torrent download without any problems.
The main basis of the torrent system is that once you are done leeching, you allow others to download from you, thus making you a "Seeder" as well. More seeds mean a faster download. Ideally, everyone wins. However, some torrents are simply downloaded, completed, and never seeded, because seeding takes up bandwidth.
Summary:
A .torrent file
IS ONLY A MAP OF HOW TO GET WHAT YOU WANT, NOT THE FILE ITSELF!
clients are what connect you to the peers who have the desired file and download the file in an organized manner.
seeder= file uploader
leecher=(obviously) the downloader
So, with that explained, go download one of these clients and begin to leech the torrent (unless nobody is seeding it....then it's considered dead)
http://pingpong-abc.sourceforge.net/ <-----I use this one
http://www.bittornado.com
http://www.bittorrent.com <-----Supposedly the tried and true original torrent app...I dunno, I don't care...
http://azureus.sourceforge.net/ <---java bittorrent client