Okay, first things first. Bittorrent is not a Kazaa replacement. Kazaa is great for individual songs and small files but bittorrent really excels when huge files are being shared. Better to keep both as you wont find individual songs on bittorrent.
Right how it works - 1 person has a file they want to share, they create a torrent (which contains information about what they want to share) and upload that torrent to a tracker. People download the torrent from the tracker and open it in a bittorrent client which then begins to connect to other people and download the actual file. That is it!
If you just set your firewall to give bittorrent complete access to the net, you should have no port problems. Seeds are people who have 100% of the file you're downloading and so are only uploading it to other people. A peer (or leech) is someone currently downloading. The Swarm is how many people are connected in total wether you are connected to them or not... Hope this makes sense :/
Also, when you have finished downloading, you become a seed. Please seed your file as long as possible so other people have a chance of getting it. thanks :D