BitTorrent is a P2P file swarming protocol, it has the ability and reliability of serving large files over the internet. It allows you to download pretty big files such as videos, movies, music, etc, in a pretty safe manner. It's the next best thing after direct downloads, and is alot safer than a direct download due to how bit-torrent files are downloaded. You get bits and pieces of the file from many other people who already have it, until you have the whole file. Kind of like putting together a puzzle.
Since it's used in P2P applications, this also means that as soon as you start downloading, you'll start sharing (uploading) what you have downloaded so far. This is why bittorent is so succesfull, they solve the problem of more customers requiring more bandwidth by letting the customers help them distribute the content.
First you need to install a BitTorrent client. We recommend using uTorrent. You can find uTorrent here.
After you have uTorrent installed, just open the torrent file you want to download and run it in uTorrent.