Well, until this is moved, I guess I will help.
The broadband adapter lets your Gamecube communicate with (1)Other Gamecubes and/or (2)a Computer.
Right now there are 3 games that can utilize the BBA. Phantasy Star Online, Kirby Air Ride, and Mario Kart: Double Dash!!. The first one, PSO, is the only official online game for the Gamecube. To play PSO online you need purchase a subscription. I'm not sure of the price but it's about $10-$15. This is an example of the Gamecube communicating with a computer. The Gamecube uses the BBA to link to Sega's servers. The other 2 games, KAR and MK: DD (and soon to be 1080 :avalanche) can use the BBA for LAN (Local Area Network) play. This is the Gamecube communicating with other Gamecubes directly. The back of the MK: DD manual shows examples of how this can be done. Basically with only 2 cubes you can use an ethernet crossover cable, and with more you need a hub.
That is all the official stuff. Unofficially you can play LAN enabled games over the internet by using a computer running a tunneling program. I was going to refer you to the thread about Warp Pipe but it seems you were the one who started that thread so I won't bother. For more info on how Warp Pipe lets you play LAN games online check their site and read the various FAQS in the forums.