Vuze, originally known as Azureus, is a BitTorrent client written in Java, which makes it rather platform independent. The Eclipse SWT toolkit is used for providing the user interface functionality.
The BitTorrent protocol is a new way of exchanging or distributing data over the internet. Downloading also means uploading, and the amounts of each are linked, to ensure fairness and rapidity in the spread of the file at hand.

For information on how to set up Azureus, see the Initial_Setup_Guide
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 client 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 is easy to customize your copy of Azureus to exactly fit your needs. This guide is targeting the Windows and Macintosh versions, although Vuze also supports other platforms.

Views

Since version 3.0.3.4 Azureus comes with a new Vuze GUI replacing the traditional Azureus application interface. The current Vuze interface:

The active windows are shown in the sidebar. The active torrents are displayed through My Library (and its subsets Downloading and Unwatched). Vuze HD Network offers a browseable limited directory of available torrents, and you can also search the open internet for new torrents. There are also separate pages for e.g. Subscriptions, Devices and for each active plugin.
The upper panel shows the downloading torrents, and the bottom panel the completed seeding torrents. New users are initially shown the "Simple view", which shows only a limited amount of information per torrent. You can change that to "Advanced view", which enables you to see more information. Contents of the most table views can be modified through Column setup dialog.
The MyTorrents view can be seen as the main view of the traditional "classic" Azureus interface. MyTorrents in the 4.2 version: