A seed is someone that has the entire file and is looking to share it with others. Primarily, in most torrents (especially if the torrent is more than a couple days old), you will be downloading from seeds.
Peers, sometimes referred to as "leechers", are people that need parts of the file to complete it (whether it's one tiny bit or the whole thing), and need to get it from others. Usually peers get it from seeds but if there are many peers or few seeds peers will share with each other in order to make the download as quick as possible.
Quite simply, the greater the number of seeds/peers, the more download sources you have, so the quicker the download. If you have more peers you will be able to upload more- important if you are altruistic or if you are on a private tracker that requires you to maintain a ratio (a .7 ratio minimum would mean that you had to upload 7/10 of a megabyte for every megabyte you downloaded).
Oh, and the # (bigger #) thing means seeds/peers you are connected to and the parentheses mean (seeds/peers total).