So you are basically sharing. There is no central server, as in regular downloading. As you might have guessed, torrents are the best option if you want to download illegal stuff. Since there is no central server, there is nowhere that the authorities can point their fingers at. So, torrents are used to download pirated stuff.
To download files using torrents, you are going to need 2 things:
1. a torrent file
Since there is no central server, there is no direct link to download the file. So, we need a torrent file. Each download has a different torrent file. You can get the torrent files from sites like thepiratebay.se, kickass.to etc. Torrents are scattered all over the internet. Even some distributions of linux's main website use torrents to serve their linux iso(although it is to reduce server load, not because its illegal).
2. a torrent downloading software
Torrents are not generally supported by ordinary download managers and browsers. So, they require a different software for downloading torrent. Some of the popular ones are Bit torrent and Utorrent. I recomment Utorrent, for reasons beyond the scope of this post.
So, once you install a torrent download software and get torrent for the file you want to download, just open the torrent file with the torrent downloading software and start downloading.