Well we all know what magnets are, but we are talking about them in context of BitTorrent here.
Magnet links are essentially links that can be opened with a browser (or more specifically the browser will prompt you to launch an appropriate third party application) that take the place once occupied by .torrent files.
Torrent files contained information about the files that are being shared, the tracker and a unique hash key that helps you to find other users who are sharing the same files.
With Magnet links, the hash information is in the link itself. There is no need for a central BitTorrent tracker as BitTorrent clients will use DHT instead.
BitTorrent websites began switching to Magnet links as opposed to hosting torrent files largely for legal reasons. While BitTorrent websites didn't host pirated content to begin with, there were always question marks around the legality of hosting a .torrent file that can be used to get copyrighted content.
With Magnet links, it is simply a string of text that is useful to a BitTorrent client, therefore it is much more difficult to prosecute a BitTorrent website for hosting magnet links. Some high profile BitTorrent sites, like the Pirate Bay, stopped using Torrent files completely in recent years in order to swtich exclusively to Magnet links.