The BitTorrent team has put together some visualizations that'll help you imagine what's happening here. The team is putting together a browser called Project Maelstrom, which can visualize torrent networks using data pulled from their popular µTorrent client and also promises to allow websites to leverage BitTorrent technology in users' actual browsers.

The team uses a bit of an odd analogy to explain torrenting actually works: you're trying to find a person (the torrent files) at the party and you're asking a bunch of partygoers (peers) where that person is. Those people point you in the right direction, leading you to the person, or people you're looking for. Instead of asking one person at a time, your torrent client can ask 50 peers every millisecond, and what you're looking for is actually some number of peers holding the pieces of that torrent you're looking for.