Bittorrent works by downloading a very small file (usually under 200 kB) from a website (like isohunt.com or mininova.org or thepiratebay.org). You then have a client program (like utorrent, Azureus, BitTornado, etc.) open that file. It will then tell the client what to download.
It connects to peers (other downloaders also known as leechers) and seeders (people who are finished downloading and are only uploading) by connecting to a tracker server that manages who's downloading/uploading from who. Everyone is always uploading because the more you upload, the more speed you will be given by the tracker. In this way, it is better than Limewire in that there are usually better sources to download from.
The advantage that Limewire has is sheer speed. You can easily get into the hundreds of kBps with Limewire, but Bittorrent will usually stay under 100 kBps unless it is a very popular torrent. On the other hand, many files that you can't find with Limewire will be readily availible on Bittorrent sites.
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