A downloads is the number of how many times the torrent has been downloaded according to its tracker record

A seeder is a peer that has a complete copy of the torrent and still offers it for upload. The more seeders there are, the better the chances are for completion of the file

A leech is usually a peer who has a negative effect on the swarm by having a very poor share ratio - in other words, downloading much more than they upload. Most leeches are users on asymmetric internet connections who do not leave their BitTorrent client open to seed the file after their download has completed. However, some leeches intentionally avoid uploading by using modified clients or excessively limiting their upload speed. The term leech, however, can be used simply to describe a peer - or any client that does not have 100% of the data

A peer is one instance of a BitTorrent client running on a computer on the Internet that other clients connect to and transfer data. Usually a peer does not have the complete file, but only parts of it, however, peer can be used to refer to any participant in the swarm (in this case, also known as a client). Note that the colloquial definition of peer is anybody, leech or seed, involved in a torrent

A health (or avaiability) the number of full copies of the file available to the client. Each seeder adds 1.0 to this number, as they have one complete copy of the file. A connected peer with a fraction of the file available adds that fraction to the availability, if no other peer has this part of the file. (ie. a peer with 65.3% of the file downloaded increases the availability by 0.653, when two peers who both have the same 50% of the file downloaded and there is one seeder the availability is 1.5)

Luba Shumeyko - Twins at Home