It's a file-sharing protocol with which users share their internet connections to exchange files. Basically someone starts
seeding a file using his normal inet connection (dedicated servers are far less common, unlike with normal downloads) and others start downloading from him. The neat thing about it is that the users not only download from the original source, but also exchange missing pieces among each other, quickly creating a huge "torrent" of data. It's usually very fast given enough people participate, but it's important that all users upload (
seed) enough themselves to contribute to the system fairly.
Leeching is when you don't
To get started you need a torrent client with which you open a .torrent file that you find on the web. Also you should make sure your ports aren't blocked by a firewall. That's basically all there is to it.
For windows users i recommend
µTorrent - has everything you need, super tiny, super low on resources.
I use torrents a lot in recent times. I prefer them to normal ftp or http downloads, because they're oftentimes faster than your average server (especially if the server gets hammered), allow you to pause/continue at any time and most clients have built in error correction to prevent corrupt downloads.