BitTorrent is a file sharing technology that allows you to download a file in pieces from many different computers located around the internet simultaneously. This is done using the software BitTorrent. A .torrent file describes to the BitTorrent software where to find the pieces of the file you want to download.
When you are searching for files to download, you first download the .torrent file, which is opened by BitTorrent. At that time, BitTorrent can use the information in the .torrent file to start downloading your file.