Bit torrent is a great way to download stuff from the internet. Here's a quick and dirty guide to using it.
A bittorrent client is a program that opens a bittorrent file and download the stuff you want. (The bittorrent file is a small file that tells your bittorrent program what files to download and how to find them)
If you use windows, I'd suggest UTorrent. I haven't used this program for years but it seems to be what all the cool pc kids are using. Hopefully it can detect your settings and works "out of the box" but you might have to follow the directions to get around firewalls etc...
If you use a mac, I'd suggest Transmission. Generally, I have found this program to, in the spirit of macintosh, "just work". There are some settings you can tweak to speed things up but they are not necessary.
Once your bittorrent program is installed, it is now just a matter of finding the torrents for the files you want, downloading them, and opening them with the bittorrent program you installed in step one. (Alternatively you can set the bittorrent client to open the file automatically)
There are a lot of places to find torrents. I use www.torrentz.com . Go to this site and search for the file you want. If you are trying this out for the first time, search for something popular like "The Beatles". A list of files will show up; click on the one you want (e.g. - "The Beatles Greatest Hits").
After you click on link, a list of all the bit torrent sites that have the file(s) will show up. Click on one of these The Piratebay or btjunkie.org are good ones.
You should now be on a specific bitorrent site (such as "thepriratebay.com") and on the correct bitorrent file page (in this case, "The Beatles Greatest Hits"). This page will have info about the files you want to downlaod. Most bitorrent sites have a confusing "download" button that isn't what you want. Rather, look for a usually smaller link that says specifically "Download this Torrent" and click on that. This will either open your bittorrent program automatically or download the bittorrent file to your computer. If it downloads to your computer (usually takes a second), find the file, right click it (on a mac press the control key and then click) and choose to "open with" the bittorrent program you downloaded above.
At this point, your bittorent program should open and show you a list of files that are about to be downloaded. In the case of the Beatles example, it would be a list of .mp3 files. There are also usually some .txt files but you don't need these. Click "add" and the files should begin downloading. This can take a while depending on the size of the files and the number of people sharing. Your bittorrent program should notify you when the files are downloaded. You may want to stop/remove the bittorent transfer once the files have finished downloading, otherwise they will continue to share them and use up your upload bandwith. (It is considered good form to share them for a little while before stopping the transfer).
If you've downloaded mp3s or other music files open them with your favorite music player such as Itunes. For any movie files, I'd highly recommend the free VLC media player.
There is a good chance I didn't communicate this very well, here are some other links to how to use bittorent: