BitTorrent in general is a file sharing protocol -- it has many uses, both legitimate (it's used widely to distribute things like Linux ISOs and other free software) and illegitimate.
Essentially, it's a peer to peer protocol -- you have a "swarm" of users who are currently downloading a file, and chunks of the file can be uploaded and/or downloaded between members of that swarm (with appropriate checksumming and authentication) until a member has the entire file, at which point they can leave the swarm (or say around and "seed" the file uploading more chunks to other people who are still downloading as a service to the community...).
The "swarm" is controlled by a "tracker", which is a small hosted program on a website that lists people currently in the swarm. Tracker sites are those you often hear about in the media being prosecuted for copyright infringement -- however, this is a legal grey area, since they're not actually hosting any content, just metadata about who does!
One of the most popular client programs used for BitTorrent downloads is called Azureus.
Azureus has two "modes", one where it acts like a traditional torrent client, and one called "Vuze", which is an effort by the Azureus authors to create and promote a legitimate video sharing community based off of licensed content. The "Vuze network" is a large repository of entirely legal video they've agreed with the producers to host, which are then distributed via a slightly modified version of the BitTorrent protocol.
Therefore, any Vuze link you've received is almost certainly legal.
Go ahead and download Vuze, not only will you be able to watch the video your friend has sent you, you'll also be able to access a world of legal, free software that will download far faster than anything centrally hosted on a private web server.
From a theoretical point of view, it's an incredibly clever and well thought-out protocol, and can be thought of as a kind if "third generation" peer to peer (after the early days where a central server was involved, and the second generation, of slightly dicey peer-to-peer where downloads were single-sourced).
It's thought that BT alone now accounts for 35% of the traffic on the Internet, although unfortunately much of this is copyright infringement.
