When you put the .torrent file you've downloaded into a BitTorrent program (client), such as uTorrent, it goes to the tracker and asks where to download from. The tracker itself has the file, so will start sending you that file. This is no different to the first I explained above. However, this tracker knows who else is downloading. So, this other person may be downloading other pieces of the file that you haven't got yet. In which case, it'll tell them to send you some stuff, and you send them some stuff that they haven't got. This cuts down on the bandwidth that the main tracker computer has to use, because the people downloading the file are helping eachother.
This can also help with speed too. You may only get a bandwidth of 1 block per second from any one person you download from, but if you're connected to 20, you'll get that 100 block file in 5 seconds.
Now, once you finish downloading the video file, you become what's known as a seed. This is just someone like you who's been downloading it and has now got the entire file. They just share to others, without getting anything else from the others. While you're downloading and sharing your pieces with others, you're a peer. You're on the same level as everyone else downloading, so you're all peers.
uTorrent will tell you how many seeds and peers there are for that particular torrent file: the more the better.
So, you need to let it download (it should say how much time and data remaining), and it'll tell you when it's complete.