The cheapest, and best CD / DVD scratch-removal compound in the world*:

Purchase a can of Brasso metal polish at Wal-Mart, hardware store, or the supermarket (can be found with household cleansers). It's cheap at about US $4.00. The cans I've seen contain about 6 fluid oz: enough polishing compound to repair hundreds of scratched discs. You only need a little bit of Brasso per disc- about two or three "moistenings" worth. It dries very quickly, so you may need to re-moisten the paper towel / tissue. As Lobsang Rampa mentioned, working from the center out, radially, is the best way, but I'm a lazy guy and tend to use a series of small circular buffing motions, gradually working my way around the disc. One disc takes me about one to five minutes to treat. Brasso leaves a slight residue, so wash the disc in soapy water afterwards; you should be doing this anyway. Dry the thing and marvel at the slightly-hazed surface- it may appear hazy to you but it's perfectly transparent to the laser, which is all we care about (any "CD repair kit" will leave the same haze).

If the scratches are deep you may want to have the disc refinished (milling and polishing the surface): just Google for this... there's a company called Auraltech that does it rather cheaply.

Cap the Brasso tightly as soon as you're done moistening the towel- my sister spilled most of a can during a particularly aggressive disc-repair session, flailing arms, etc.

For future reference: you may have to fight the urge, but for Bob's sake if you're going to set a bare disc on a counter, desktop, etc. do NOT set it label-side-down, shiny-side-up. You will be punished for your ignorance. Any grit on said surface will easily scratch through the whisper-thin layer of lacquer and ink on the label side of the disc, and tear up your aluminum substate... and this is the side on which your data are stored. Once the substrate has been scratched, you're fucked. Now, if you set your disc shiny side down, counterintuitively, I know, if it is scratched, at least you can polish the scratches out with Brasso. Think about it: would you rather be polishing a piece of plastic, which can take scratches up to a millimeter deep before they touch your data, or reapplying a several-molecules-thick layer of aluminum?

* Brasso scored highest out of nine cleaning compounds, including commercial CD repair kits, AND toothpaste, which proved to be useless, in an article entitled "CD Repair Kits" on a site called "Burning Issues." They have before / after waveforms to prove it: quantifiable data for the whole family!