When you d/l a .torrent file, you're downloading something that tells your client (uTorrent or Azereus) what file you want. When you double click on the torrent, you're telling the client that you want to d/l now, and then the client will look for other computers (peers/seeders/leechers) from which to d/l from. When it tells you that you're not connected to a peer, it means that someone isn't online to download from, so it should try to connect later.
Torrents work so that you don't have to download the entire file before you can start uploading it, like say in limewire or frostwire. This means that before you even finish downloading the file, you'll be sharing with others. This is good because if you get dis-connected, once you're reconnected, you can still download w/o loosing progress. You also don't necessarily need too many seeders, just one should be good (but more means faster).
Peers=people just like you, who are in the process of downloading
Seeders=people who've already downloaded the file and are allowing other clients to access their file so it can be shared
Leechers=people who haven't downloaded much of the file yet and so have nothing to contribute to other people
When downloading a torrent, you should check to see how active it is, you want to make sure it has at least one seeder.