Torrent

.torrent files are basically BitTorrent's equivalent to ed2k links used in eDonkey network. In essence, .torrent files include instructions for BitTorrent -compatible P2P clients where to find the "tracker" for the downloaded file, hashed checksum that verifies for the client that it is actually downloading the correct file, the full size of the file and the filename of the file.

So, essentially .torrent files are very, very small files that simply contain some textual data about the file itself. Thus, downloading the torrent file itself takes only matter of seconds, if not milliseconds, but it is not the file that is downloaded from the P2P network. Instead, .torrent files are typically hosted on just standard web servers and can be even passed via email. But in a computer with a BitTorrent -compatible client installed in it, launching a .torrent file typically starts a process where the BitTorrent -compatible client first contacts the tracker described in the .torrent file and if the tracker server is found and "seeds" (people sharing the file in question) are available, launches a download process of the file described in the .torrent file.

To use .torrent files in order to download files from BitTorrent P2P network, you need to have a BitTorrent -compatible client installed on your computer. Some of the clients are available here:

BitTorrent (the original client, open source)
ABC (open source)
Azureus (open source)
BitComet (open source)
BitTornado (open source)
G3 Torrent (open source)

If you are interested to know more about the BitTorrent, you might find these links useful:

Information about BitTorrent itself
BitTorrent FAQ
ABC guide
Azureus guide
BitComet guide
BitTornado guide
G3 Torrent guide
AfterDawn's P2P forum