A .torrent file is a file that contains the basic information about a file or set of files. This includes the file names, sizes, the date created and some other information. The torrent file also contains an info hash - which is basically just a unique code for that torrent. No other set of file(s) will have the same info hash as this one. The final thing contained inside the torrent is a list of trackers. If you have some torrent files, you can't open them in a text editor because they are encoded. You will need to open them in torrent client software (like µTorrent which you can install on your computer.
When u r searching for the file, u r said to be a peer. So what happens here is that u download a torrent file (containing info of a certain file u want to collect). Then u open it using a torrent client. This makes us start downloading the file, in pieces, from different seeders. A seeder is a person who has more than 0% of the files described by the torrent. So anyone downloading the file is a seeder, including u.
Why is it done? To utilize bandwidth effectively. Supposing many people(seeders) r distributing the same file, more exchange of data occurs, so everyone gets to download it as fast as their download rates allow them to.
The catch... however, is that u've to upload the file too... to allow it to propagate. If u don't upload, u r a bad seeder. An unspoken rule which tells that u r not a good propagator. Think in this terms... supposing everyone like u don't propagate the file, there wont be any file to download!!!
When u start torrenting, just remember that u can put download to the max bandwidth, if possible, but an upload of @least 10kBps'll be helpful to others too.