BitTorrent uses its own special set of buzzwords to describe various aspects of how the system works. If you want to really understand what's happening, it helps to understand what the terms mean.
Seed: This is the machine that originally distributes the shared file. Technically, any peer that has 100% of the shared file also becomes a seed.
Peer: This is a machine that downloads the shared file and then shares parts of the file to other peers. When you run a BitTorrent client on your machine to get a file, you're running a peer.
Piece: Part of a shared file, to be combined with other pieces when reassembled into a usable whole.
Torrent: Torrent is used in two ways. When specified with the word file (as in "torrent file"), this is a file containing metadata describing all of the pieces of the shared file and its checksum (validation) data. When used on its own (as in "I downloaded a torrent of Ubuntu"), it means the shared file (and all its pieces).
Swarm: This is the full network that's sharing a file, consisting of all the peers and seeds.
Tracker: This is a server that keeps track of the seeds and peers in the swarm. A tracker is often not involved in the actual transfer of data, but acts as more of an index or search engine for people looking for torrents. Trackers are often the target of legal action, because they're seen as the enablers of illicit file sharing.
Leech: This is a term for a peer that does not share pieces of a shared file. Basically, this is someone who wants to download a file using BitTorrent, but is not willing to do their part to support the swarm. Some leeches on very low-bandwidth connections can't download and share, so they download first, then share. But most leeches simply choose to be a "bit piggy" and download files without giving back.
Health: Do not confuse the BitTorrent term "health" with quality or safety of a file. In BitTorrent, health means how much of a file is available to download (anything less than 100% health means you're not getting a full file today). Do not assume something that shows 100% health is, for example, free of viruses. This also spotlights one of the downfalls of torrenting: Not all files are fully available to download. You may have to wait until a peer shows up with the missing pieces or, for less popular files, you may never get those missing pieces.
Fake: A file that spoofs what it claims to have. Generally, it's a file that contains malware or just junk bytes, designed to either attack users or improve the uploader's ratio.
Ratio: This is a measure of how much you've uploaded. It is sometimes used by trackers to allow more access to more files.
TOR: Folks often get confused by this. TOR stands for The Onion Router and is unrelated to BitTorrent. Because TOR is a way of communicating over the Internet anonymously, some people do run torrents over TOR. But the TOR Project and BitTorrent are completely separate beasts.
Have I missed any important terms for torrents? If I did, let us know in the comments below.