A torrent is a file sent via the BitTorrent protocol. It can be just about any type of file, such as a movie, song, game, or application. During the transmission, the file is incomplete and therefore is referred to as a torrent. Torrent downloads that have been paused or stopped cannot be opened as regular files, since they do not contain all the necessary data. However, they can often be resumed using a BitTorrent client, as long as the file is available from another server.
Torrents are different from regular downloads in that they are usually downloaded from more than one server at a time. The BitTorrent protocol uses multiple computers to transfer a single file, thereby reducing the bandwidth required by each server. When a torrent download is started, the BitTorrent system locates multiple computers with the file and downloads different parts of the file from each computer. Likewise, when sending a torrent, the server may send the file to multiple computers before it reaches the recipient. The result is a lower average bandwidth usage, which speeds up file transfers.
Just adding a bit of detail.
To share a file or a group of files, people create *.torrent files. These files contain data about the files being shared, and about the trackers (computers which coordinate the file sharing). So, even if you have deleted the *.torrent file (or have incomplete files from some other computer), but can get the original *.torrent file from somewhere, you can resume downloading the incomplete files.
[Note (If you're fairly familiar with the above): Most new BitTorrent clients also support "tracker-less" torrents, which use DHT's to coordinate file-sharing. While this is not as fast as using a true tracker, it eliminates the need for the said tracker]
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