Baking soda toothpaste
Backing soda toothpaste, huh? Actually, any mild abrasive like furniture polish, Pledge, or plastic polish works fine for removing CD scratches. Just take any lint-free cloth, add some of the abrasive to the afflicted area, and then wipe. Make sure to wipe from the center of the disc to the rim in a straight line. Never wipe in a circular pattern.

This only works on minor scratches, but I found the results were OK. While it does not work well in bringing back heavily scratched CDs from the grave, it does help to stop CDs from skipping with light to moderate scratches on them.
Finally, use baking soda toothpaste with as few additives as possible. Some of the gel-based baking soda toothpastes tend to leave a sticky film on the CD, which requires a bit of effort to remove.You can use furniture wax like Pledge to remove the film.

When cleaning your CDs use a soft, lint-free cloth and make a single swiping motion from the center of the disc to its rim. Don't clean in circles like you would a vinyl LP.
Remember, the worst scratches are the ones that go circularly around the disc. A scratch from the center to the rim isn't as bad. A CD player can miss a beat and you won't notice it, but if the scratch follows the track pattern of a CD you'll notice a lot more skipping.

Finally, a word of caution: The worst place to scratch a CD isn't on the bottom but actually the top. Why? Because the label side contains the reflective material required to bounce the laser back to the CD player's pick up head so it can be played correctly. It's also close to where the pits and bumps that make up the data track on a CD are stored. Scratch that and you'll most likely never play the disc ever again.