This is the number of copies of a torrent that the client can detect. Because every seed has a complete copy, they will always add 1.0 to this number. A peer with a small portion of the file downloaded can only potentially add a fraction to the availability, but only if they are contributing to the completion of a whole copy.
For example, if a peer has downloaded 32.3% of a file, they can only increase the availability by a maximum of 0.323. If two peers have downloaded the exact same 32.3% of a file, they will still only add 0.323 to the availability. But if the second peer had a completely different 32.3%, then the increase in availability would be 0.646.
Choked
If a client refuses to share with another client, then it is described as being choked. This can happen if the second client is a seed, and doesn't require any more pieces, or if the client is uploading at full capacity and cannot accommodate any more connections.
Interested
A client will flag a peer as 'Interested' if it detects that the peer is missing a portion of the file that the client possesses.
Leech
A leech is another name for a peer who uploads much less than they download, resulting in a poor 'share ratio'. This will often happen if the client is not left to seed the file for a while, after the download has completed.
Optimistic unchoking
Every 30 seconds or so, the client will update its list of uploaders and try to reconnect to those that were previously choked.
Remote connection
This describes a peer that has connected to your client.
Reseed
If at any time a torrent download has no seeds, and too few peers to make up a full copy, you will be unable to complete the download. To fill in the missing pieces, it is necessary for a seed to rejoin the swarm and allow the peers' downloads to finish. This is called a reseed.
Scrape
A scrape is a client's request for statistical information from a tracker.
Share Ratio
This is the amount of a file you have uploaded, divided by the amount you have downloaded. The result shown is representative of the current session only (since you last opened your client), and not the whole time you have been downloading the file. A share ratio of 1.0 means you have uploaded the same amount as you have downloaded.
Snubbed
A downloading client will describe an uploading client as snubbed if it has failed to download any data from it for more than 60 seconds.
Superseed
Most BitTorrent clients now have a superseed mode which should only be used by the initial seeder, before a significant swarm has been formed. This is to prevent the seed from sending out the same pieces of the file to the relatively few peers that it has, causing low availability. Instead, the seed will only send out the pieces that haven't been sent before, increasing availability and speeding up file propagation.
Swarm
This is the sum total of every client sharing a particular torrent, made up of the seeds and the peers.
