Pokemon Box is a Gamecube game that is nothing more than a utility for storing your Pokemon. It’s sort of like the boxes on your PC in the gameboy game, except you can store a lot more Pokemon (up to 1500 Pokemon total!).
You can also use Pokemon Box to play your gameboy game on the TV.
Oh yeah, and you can get eggs containing Pokemon with special moves if you store a certain amount of Pokemon:
Swablu with False Swipe, Skitty with Pay Day, Zigzagoon with Extremespeed, and Pichu with Surf.