BitTorrent is a peer-to-peer file sharing (P2P) communications protocol. BitTorrent is a method of distributing large amounts of data widely without the original distributor incurring the entire costs of hardware, hosting and bandwidth resources. Instead, when data is distributed using the BitTorrent protocol, each recipient supplies pieces of the data to newer recipients, reducing the cost and burden on any given individual source, providing redundancy against system problems, and reducing dependence on the original distributor. The protocol is the brain child of programmer Bram Cohen, who designed it in April 2001 and released a first implementation on 2 July 2001. It is now maintained by Cohen's company BitTorrent, Inc.¹
Usage of the protocol accounts for significant traffic on the Internet, but the precise amount has proven difficult to measure.
There are numerous compatible BitTorrent clients, written in a variety of programming languages, and running on a variety of computing platforms.

To be able to download a file, you first need to get the associated .torrent file. This file, usually a dozen KB in size, is the “signature” of the much bigger file to be downloaded, and it needs special software to be read properly. Azureus is one of those types of software.

If you want to host files yourself, you need a tracker, which is basically a central server coordinating the connections between peers. Azureus can provide a tracker too.

You will find, as you discover its functionality, that it easy to customize your copy of Azureus to exactly fit your needs.

Torrent files can be downloaded from all over the place. And lots of companies release demos of games and movie trailers via torrent. Try these -

uTorrent for Windows
Transmission for Mac
Azureus or KTorrent for Linux (Azureus is cross platform, meaning it will work on Windows and Mac)
Mininova
The Pirate Bay
isoHunt
TorrentSpy
btjunkie
Demonoid