How do BitTorrent magnet links work?

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On a high level, magnet links let you download the torrent file via BitTorrent. They encode the infohash and some more information. The infohash makes it possible to uniquely identify the torrent. BitTorrent clients use information in the magnet link to get the same info that is contained in the torrent file from peers.

Let us consider the following magnet link, captured in the wild:

magnet:?xt=urn:btih:dc2e7bf4a273dc4b25ae96e833fd50be2b00e953&dn=The+Humble+Indie+Bundle+6+for+Windows+%2B+Soundtracks&tr=udp%3A%2F%2Ftracker.openbittorrent.com%3A80

Let us look at its anatomy:
 

• magnet: — URI schema; this tells the browser that this is a magnet link and, if the browser is configured correctly, invokes a BitTorrent client to deal with it
• xt=urn:btih:dc2e7bf4a273dc4b25ae96e833fd50be2b00e953 — the meat; this says that this is a BitTorrent infohash type of magnet link and gives the value of the hash; technically, many different ways to address content by a cryptographic hash are supported, but the BitTorrent infohash is the only type of magnet links currently in wide use
• dn=The+Humble+Indie+Bundle+6+for+Windows+%2B+Soundtracks — the (URL-encoded) name of the download, so that it can be displayed before this information becomes available from the peers
• tr=udp%3A%2F%2Ftracker.openbittorrent.com%3A80 — the (URL-encoded) tracker; this helps to find peers that are participating in the torrent

Once the client gets the information in the torrent file, it downloads as usual.

The tracker part is commonly used, but not required. Presence or absence of trackers is orthogonal to magnet links vs torrent files. Regardless of whether the client has the torrent file contents, it can find peers using the distributed hash table instead of trackers. In both cases, peer exchange is the main mechanism to find peers after finding the first one or two with DHT or a tracker.