Torrents in a nutshell:
You download some software that sets up your personal computer as both a client and a server.
A central computer keeps track of everybody who WANTS a certain file and everybody who HAS that same file.
The software you've started on your own PC will download part of that file while simultaneously uploading to any number of other people.
IOW: As you download portions of the whole file, those portions on your computer get served to others.
Eventually, in the best case scenario, everybody gets the entire file.