BitTorrent can be difficult to understand so I'll try to explain how it works. Let me know if there are any mistakes.
BitTorrent is NOT Napster
BitTorrent is not a file sharing technology. It's a download sharing technology. With BitTorrent, you share with others what you are downloading while you are downloading.
What is a Torrent?
A torrent is just a web resource served by a web server and accessible by a URL. Usually a torrent maps to a file or a folder of files. Link to a torrent can be published and shared like other links so torrents are blogosphere-friendly.
How do I download a Torrent?
To download a torrent, you need to install a BitTorrent client that will run when a torrent link is clicked on. Note that there is no need to launch the client explicitly so don't be surprised if there is nothing to launch after installing the client.
While Downloading a Torrent…
As mentioned above, clicking on a torrent link will cause the BitTorrent client to start downloading the torrent. Torrent-enabled web server keeps track of the downloaders and refer them to each other so they can share what they downloaded so far. If there is just one downloader, there is no one to share with so it's just regular file download.
If there are many downloaders, some of the downloaders will have parts others don't and could share those parts with others. This reduces load on the web server and allows it to handle large number of downloaders easily. Bandwidth saving also means a shocking ISP bill won't wipe the slashdotted smile off your face at the end of the month.
While a torrent is being downloaded, some (all?) BitTorrent clients will scramble the content for reasons I am not sure of. If there are more than one file in a torrent, all the downloaded files will be scrambled until all the files are downloaded completely.
Where Do I Find Torrents?
Websites, blogs, and Google. There is no central directory like Napster.
In my opinion, flash flood nature of blogs will be well served by BitTorrent. Likewise, link-happy nature of blogs will complement BitTorrent well. Ultimately, I think a tailored variation of BitTorrent should be built into blog clients and servers for download sharing of feeds, images, enclosures, and other blog-related resources. BitTorrent will encourage media-rich blog posts without applying power-law to the bloggers' wallet. BitTorrent means blog torrents. [Don Park's Daily Habit]