Note that SOME torrent downloads can be legal - Linux and OpenOffice are "free" for example.
Basically when a person makes a file available on a torrent site they are the first "seeder" as they are the only one with a complete copy.
As other people start to download it they become "leechers" (I guess also known as peers) as they dont have the complete file. But even though they dont have the complete file other people can download it (parts of it) from them.
As more and more people get the complete file the number of seeders will go up. As more and more people request it (but dont have the complete file yet) the number of leechers (peers) will go up.
Generally the more seeders (and leechers) available the quicker you will get it as you download a bit of it from each person and join it together on your PC.
Note that YOUR download speed also depends on a persons UPLOAD speed (the person you are getting it from). If THERE upload speed is slow then YOUR download speed will be slow.
Note also if they turn their computer off you wont be able to download it at all. Note also if they download it, but then delete it from their PC, you wont be able to download it either.
Generally the more seeders (and leechers) a file has the quicker you will get it.