I am everywhere, you just can't see me, I just make myself visible when I'm summoned, but actually I've been there all the time ;-P
I really do think David could explain this better than me, but here you go:
As you know, Bob, files are usually uploaded somewhere on the net (meaning on some provider's hard disk somewhere you can connect to, to simplify to the max degree). Torrent files, however, are shared by the users directly. That means that when you download a file using this method, you are actually connecting directly to the computers of other people who give you access to a specific folder within their own hard disk. This system has the ability to look for all the computers that are connected to the internet at a specific time and pick up pieces of identical files from all of them at the same time. That's why it is referred to as "peer-to-peer" sharing, there is no intermediary in between the users.
Obviously, that does not mean that every computer is part of this network, but only those computers that have a specific program installed that will allow them to do this.
In other words:
let's say Jim uploads a show to his provider's server, you connect to it and download the show.
However, let's say that Jim hasn't uploaded that show, but Peter, Barry, Meow and I all have it on our hard disks because we shared it some time before as a torrent. Let's say you have the torrent sharing program installed as well, so you can download the file directly from us. At first only Peter and Barry are connected, so the torrent software starts picking up pieces of the file from their computers, then Peter turns off his computer, but I connect to the net, so the program "sees" my computer and starts taking pieces of the file from my computer as well, and so on. Except for Meow, because she's on vacation in Egypt and is busy trying to buy the pyramids for her garden. If nobody is connected, the file can't be downloaded, you have to wait until somebody turns on their computer.
Not sure how clear all this was, but I hope it helped a little...