Today's game systems use discs, which can not be used to save your game data
like carts were back then, so you have to have another way to save.  That's
what memory cards are for.  They're basically little cartridges, but they
store your save data instead of the actual game and your saves.
That's all you need, though--a Gamecube, a game, and a memory card.  The
system comes with the power and connector cables, and a controller, but just
one.  Actually, I think there is a Gamecube bundle with Mario Kart and two
controllers inside for $99 at the moment...and there's a bundle with Metroid
Prime in it, too, which is a great game.