Torrent or BitTorrent is a peer-to-peer file-distribution protocol that lets you download large files without putting too much strain on one single server, because the download is distributed.
Although frequently referred to as "downloading", the BitTorrent protocol relies on all users to be recipients and distributors of .torrent files. While you download a BitTorrent archive to your computer, you are also helping to distribute it to other users. It is not possible to download without uploading, although you can of course set restrictions on bandwidth usage.
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.