Wash the disc in warm water and soap. Dry it carefully with a soft cloth (a cloth diaper works very well). Even if the disc appears very scratched, skipping is often caused by dirt and fingerprints and not the scratches.
Put a drop or two of a very mildly abrasive metal polish onto the scratch. Rub it very gently with the cloth. Try to rub in straight lines from the center of the disc outwards. Do not rub in circular motions or around the disc. The scratch should start to fade.
Rinse the polish off under the tap. Dry it with the soft cloth.
Put a little car wax over the surface of the clean, dried disc. Let it dry and wipe the wax away.
Test the disc to see if it plays. Repeat Steps 2 through 4 if the disc still does not play or if there are other scratches.