It's the "component" cables that you can only get from nintendo.com, because Nintendo was the only company that made them for the GameCube, and they didn't think people wanted them. They're "higher end" than S-video.
You can get the official "Nintendo" brand S-video cable from nintendo.com. It's
right here. But at $16 it seems a little pricey.
You can get third party cables from a lot of different places. As an example, here's one from
lik-sang.com that's only $4.
If you look at the picture on Lik-Sang (click it to make it bigger), the big square plug on the left goes into your GameCube, the red and white plugs go into the "audio input" on your TV, and the round black plug on the right is the "S-video" part. It plugs into your new TV's S-video port. It has a bunch of metal "pins" inside it, and looks a lot like the port that your computer's keyboard might be plugged in with, but it's not the same.
Now that you know what it looks like, you might want to see if you can find one locally, in a videogame store or something, so you won't have to pay shipping costs.
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And does anyone know if I need a new cable for my PS2 'slim'? Or was it included with the system?
The head that connects to a Sony console is different from one that connects to a Nintendo console. Or a Microsoft console. So they all need their own cables. You can swap them between a PSone and a PS2 (any version) though. Or an N64/GameCube. As long as it's the same "family", it's usually okay. And no, they're almost never included.
BTW, does your TV have multiple S-video input ports? If it doesn't, you might want to look for an S-video switcher box, unless you want to manually connect and disconnect cables back and forth a lot, or just leave one console with the S-video, and the other with ordinary A/V cables.
EDIT: IT TOOK ME FOUR HOURS TO WRITE THIS, AND TWO HOURS TO READ Pro666's RESPONSE... dur hur hurr.