The Gamecube is different from the NES and N64 in that the games come on CD's instead of cartridges. Inside many of the cartridge based games are small batteries similar to a watch battery. These batteries can hold a saved game on them for many years without needing an additional memory card. Since the Gamecube plays CD's there is no way to put a battery in the CD so all saving is done through an external memory card that gets plugged into the front of the console near the controller ports. While you can still play your games without having a memory card, it's not recommended as you'll have to start your games over each time you play.
These memory cards come in a variety of storage capacities. They range in sizes from 4 MB to 128 MB. I would only recommend that smallest size if you only play 1 game as there's not enough space to save multiple files. I feel that the 64MB size is optimal because you get plenty of space at a reasonable price.
Another little bonus is that these memory cards will also work on Gamecube compatible Wii's and allow you to save your progress in the Gamecube games you play there.
The memory cards also come in original or generic styles. Originals are the ones that were made by Nintendo and have their logo on them. Generics were made by a 3rd party but do the same thing. While it is often the case that original product is superior to the generic counterpart, I haven't been able to find any problems with generic memory cards.
The one below is the one I found works best for me.
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