Wow, these answers are all terrible. I can't believe that so many people managed to all sort of write about this without really making any real kind of sense. Okay, so here we go with the real deal:
The first thing to know is that terms like BitTorrent and torrent are sometimes used for multiple things and can be interchangeable. This can get very confusing. But we'll come back to that.
Okay, so what is this BitTorrent thing? Well, it's a means of downloading (and uploading) files. Ultimately, that's all it is - it's how it does it that gets complicated.
Normally, when you download something, you click a link on a website or something and the file is downloaded from a server somewhere onto your computer. There are just two computers involved: your computer and the server where the file is. The problem with this is that bandwidth costs money. The space on the server and the connection from the server to you both costs money. BitTorrent does away with all this.
The best way to explain how it works is by way of an example. Bobby has a video file of some naughty ladies enjoying each other. He wants to make it available for other people. Isn't he nice? The file is 100MB (megabytes) in size. Let's say that ten people want to download it. Normally, this would involve them each downloading the full 100MB file from a server. That would use 1GB (gigabyte) of sever bandwidth - 10 x 100MB downloads. Okay?
With BitTorrent, this file is split up into sections. For the sake of this example, I'm going to simplify it a little, but this is still how it works. Bobby uploads a torrent file (it doesn't matter what this is yet, I'll come back to it). Person number one connects directly to Bobby's computer and begins to download. He gets the first 10MB of the 100MB file from Bobby. Person number two connects directly to Bobby's computer and he gets the next 10MB of the file from Bobby. Now this is where the magic happens. Bobby has the whole file. Person 1 has the first 10MB section. Person 2 has the next 10MB section. Instead of Person 1 getting the second section of the file from Bobby, he gets it from Person 2. And instead of Person 2 getting the first section of the file from Bobby, he gets it from Person 1. With me?
The file is split into ten sections. Everyone who wants to download the file is connected to each other instead of to a server. When Person 1 has finished downloading the first section of the file, it is available to everyone else that is downloading too. So in the group of people downloading, there are now two copies of the first 10MB chunk of the file. If Person 2 gets the next 10MB chunk and Person 3 gets the next 10MB chunk and so on, you'll have a situation where only Bobby has a complete copy of the file, but the ten people downloading each have a 10MB section of that file. None of the downloaders have a complete copy of the file, but between them, they do. If you were to take all the chunks of the video file that they have between them and stick them together, then you would have a complete copy of the naughty ladies video file. And that's the beuty of BitTorrent.
The ten people now share the bits they have between each other. We don't even need Bobby any more. He can switch off his computer. The other ten will pass the chunks they've got around each other until they all have a complete copy of the file. Person 1 has the first chunk and he'll send it to Person 2. Person 2 has the second chunk and he'll send it to Person 1. Between them, they will now both have the first two chunks of the file. With all ten people doing this, all ten chunks will be sent to everyone and everyone will have a complete copy of the file.
And that's how BitTorrent works. Why it's good is that a server wasn't involved. Do you remember how I was saying that normally it would require 1GB of server bandwidth for ten people to download? Well in this case, Bobby has allowed ten people to connect to him and they have each dowloaded a 10MB chunk of the file from him. In total, they have downloaded 10 x 10MB = 100MB. And they have each downloaded 10 x 10MB = 100MB from each other. So instead of Bobby using 1GB of bandwidth, he has used 100MB and so has everyone else. The burden of downloading has been shared between them.
For all this to happen, some software is required and a special type of file is also needed. To download, you need a BitTorrent client. There are many of them out there. The other thing you need is a torrent file. The torrent file is created by Bobby. It contains information about the naughty ladies video file. He uploads it to a tracker (a server). The ten people who want to enjoy the naughty ladies download this file. It's very small. As well as containing the information about the video, the torrent file contains the server's address. When the ten downloaders open this torrent file in their BitTorrent client, their computer will contact the tracker to tell it that they are downloading and to