**DISCLAIMER** Do not use torrent apps to download warez/pirated stuff. **/DISCLAIMER**
First, check out
µtorrent, it's a good light-weight torrent client.
Second, be sure the sites you are getting your torrent files from show the seed/leech count and allow users to comment on the files. Many bad files (viruses, phony music/videos, bad data, et cetera) will be noted by regular users if the site allows it. It's a good way to judge which torrents are worth getting.
Otherwise, you have it pretty much right. You download a .torrent file, which basically has a marker for the file you want and a list of trackers that keep tabs on what IP address has what files available. When you load the file on your bt client, it goes out to those trackers with the file you requested and starts downloading from IPs that it knows are connected and have it available. Once you've downloaded a certain amount, you then begin seeding your completed portions while you download the rest.
FYI, it's considered in good taste to at least seed the file until you have uploaded as much as you downloaded (MB for MB, GB for GB) or more. Otherwise you are a "leecher" and it's frowned upon.