CD Repair?

Seriously - toothpaste, rinse with water. And don't forget to go from the inner hole outwards, not circular.

I bought a Disc Doctor off of Ebay a few years ago. It is essentially as the others have descrbed: a crcular platform to hold the disc and a soft cotton pad to polish out the scuffs. You apply a cleaning solution and it will buff out the problems.

I've also seen some serious commercial CD polishing machines.
Doesn't make sense for most of us but I think these services and DVD rental chains may be using them.

I've used a good quality metal polish like "Maas" to polish out some nasty scuffs and not too deep scratches from abused library cd's. It works quite well as long as things aren't too far gone. It's a bit faster than toothpaste, which does work as a good polish for all manner of things.

Sometimes work, sometimes not. I have tried that service, and the success depends on the conditions of your disc.