BitTorrent is a protocol (a set of rules and description of how to do things) allowing you to download files quickly by allowing people downloading the file to upload (distribute) parts of it at the same time. BitTorrent is often used for distribution of very large files, very popular files and files available for free, as it is a lot cheaper, faster and more efficient to distribute files using BitTorrent than a regular download.
BitTorrent Mainline is a client. A 'client' in this case is a computer program that follows the rules of a protocol. For example, HTTP (HyperText Transfer Protocol) is the protocol used to download web pages and other content - like this page - and your HTTP client (or browser) is the program you use to get those web pages. Some popular browsers include Microsoft Internet Explorer, Mozilla Firefox, Safari, and Opera. To an extent, they all work the same way because they follow the same set of rules. The BitTorrent Mainline client will give you access to the world of content on the protocol in a lightweight, fast and reliable package.