I've seen people requesting invites to torrent sites. What is that all about? How are those sites different that kazaa or morpheus or any napster-like site?

The difference between Bittorrent and other p2p softwares is that it uses torrent sites to keep track of what files are shared. You download the torrents(files with .torrent extension) of your choice from the trackers on these and open it with a bittorrent client, and the download begins. Alternatively you could use a torrent search engine like torrentspy or isohunt.
The good thing about it is that you are able to download even from users who are currently downloading the file (i.e from incomplete downloads) which ensures plenty of sources , esp. for large ones like movies.
If I'm failing to get the idea across, you could just google it

Hell! I'm stupid, too!

The good thing also is that you get a lot more speed and no queues.

So can you play the files on mp3 players? Do the files stay as a ".toorrent" files?

You load the torrent in your Bittorrent (or any other program) and then it loads the .mp3, .mpeg or whatever file. The torrent file is only used to indentify the file you are downloading.

One thing that confuses me is, do you download the pieces of a file at random or in sequential order?

Here, this will help: http://www.everything2.com/index.pl?node_id=1419083