Basically, the bittorrent protocol works like this:
You have a file (usually, but not always, a large file) that you wish to share with the rest of the world. Ordinarily, people would connect to your server and download directly, as in, click the download link, "Save File As...", etc. But there's a problem: five thousand people are trying to connect all at once to download this several-hundred-megabyte file.
Moses in the bullrushes! What's a girl to do?
Well, dear, that's where bittorrent comes in. Instead of uploading the file to a server, you upload a tiny scrap of information to something called a "tracker". The tracker serves the tiny little file, which points to
your PC! The file, still on your hard drive, is downloaded directly from your PC in small chunks using a bittorrent client.
It doesn't end there, though. Once someone is done downloading a chunk from you, it is then shared to other people downloading the file (called the "swarm"). Further, those people can stop downloading at any time, and the partial file will sit on their hard drive, waiting to be completed. When they open the client software and restart the download, it continues right where it left off, so long as they still have the .torrent file.
There are a few things to be aware of. First and foremost, be advised that it is a violation of the Digital Millennium Copyright Act to download copyrighted material in any form, by any method, if one does not have permission to do so (kind of a joke, since such permission is almost never given in the first place). Such downloads can contain viruses and/or trojans; it is best to scan each file before you open it.
Secondly, an improperly configured bittorrent client can use a great deal of bandwidth and slow the rest of your internet experience to a crawl. Check your internet provider's rules to see if you have a monthly download cap, or if their rules specifically forbid use of the bittorrent protocol.
Finally, there are malicious entities out there who scan bittorrent seeds and peers (the "swarm") for infringing material. Learn about how to use an IP blocklist within your client of choice, and learn how to apply one. I recommend the blocklists available from Bluetack; utilizing these will filter out those hosts and other bad peers. Note that this is not an absolute protection from such activities.
There are many, many bittorrent clients out there, all of which handle the .torrent file. One I have is called "
Vuze", which is an iTunes-like interface specifically for bittorrent. Vuze has a library page that has some very interesting stuff on it, and even has television and film studio partners; Vuze may be a good one to start with.
Once you're comfortable with using torrents, you may want a client with less memory overhead.
uTorrent is a good one for that purpose.
Hope I clarified some things, and happy torrenting!
edit: I forgot to add that you'll need to find what's called a torrent tracker. Sadly, these are often shut down as fast as they spring up due to copyright complaints, but two I have had luck with are btjunkie and demonoid.