Utorrent is a program uses bit torrent technology, which is a way of sharing files sometimes called peer-to-peer. In a bit torrent set-up, many people can upload and download a particular file simultaneously. This means that you can often download a file at a much quicker pace than you could with a straight file-sharing program like Kazaa, since that type of program usually only involves two people.
A bit torrent is a file that links you to a torrent tracker. You can think of it like a signpost that shows you where your file will be. Many other people can all also download the same file as you at the same time. Each one of them is given different pieces of the file at different times, which they in turn pass onto you. And as part of the torrent, you also pass pieces of this file to other people. So by making everyone in the swarm help with the file, your download rate can be greatly increased.
It's important to make sure that whatever you download off a torrent site is legal material. There are torrents on many sites that violate copyright and aren't legal for you to download. On the other hand, it is possible to find legal content such as albums that bands willingly put up for free download in order to get exposure. It pays to do a little research before you download.