Torrents started off as a way for people to legally share large files. For instance you work from home and create a large PSD file, you cant send it by email as its too big, so you create and upload a torrent and sent the link to your clients for download.
Obviously now its more used for the illegal downloading of films and software. As you are downloading a collection of many smaller files from many users you get enhanced download speeds, and the ability to pause the download and continue at a later date.
It is good form to continue 'seeding' the download once it has completed so that others can download it from you, allowing even faster download speeds.
Be warned however, there are a lot of virus's in the torrent community which can adversly affect your PC.
Two of the most popular torrent sites are. http://www.mininova.org and http://www.thepiratebay.org.
The pirate bay is probably not deemed safe for work or school as it does contain some adult banner ads.