A .torrent file is nothing more than a precise description of another type of file. For example, I could have an MP3 of Michael Jackson’s “Thriller” and create a .torrent file for it. The .torrent file contains information on the name, exact size, location of the file on my computer, etc.
I can take this .torrent file and upload it to a bittorrent tracker (a server that hosts only .torrent files, not actual files). Other users can choose to download that .torrent file with a bittorrent client (application used for opening/creating/downloading torrents), like Azureus/Vuze, uTorrent, etc. When other users grab my .torrent file from the tracker, their bittorrent clients know exactly what file to look for and where it is located, to start downloading it.
The reason bittorrent works so well is that you are not downloading from a central location, like a server on the internet. You are downloading files directly from other people’s computers. The .torrent file adds a level of ease in downloading because clients will not download the wrong files and can actually grab chunks of the file from hundreds of users at a time, then assemble the whole file on the destination computer. This allows for very fast downloading.