There are plenty of tips found online for how to clean and fix CDs, some debunked and some debunking the debunkers (oy!). Here is a list of three methods (light, moderate and extreme) that should cover all the bases.
First Step: Getting It Clean
Wash the discs with your fingers using a little detergent or rubbing alcohol, don’t use a sponge since it can mar the surface. Rinse with water, shake off and allow dry.
Once it’s completely dry, try playing it. If it works, you’re in luck! If not, move on to the methods below.
- For light damage: Apply a bit of toothpaste to a q-tip (cotton swab) and gently polish the scratches with toothpaste until you have removed them. Now wash in some water to remove the toothpaste. Dry completely then test it. If it doesn’t work, you can try the toothpaste method again or move on to the next method below. See: Hardware Secrets (Toothpaste Method) 
- For moderate to heavy damage: If toothpaste doesn’t work, try using the metal polish “Brasso”. Working in a well ventilated area, pour a bit of Brasso on a paper towel and rub the surface with the Brasso in a circular motion, applying pressure if needed to work out the scratches (careful not to break the disc). Once they’re removed, apply a light pressure polish. During the process take a few breaks to allow the Brasso to haze and then remove it with toilet paper, this helps the polishing process. Rinse and allow to dry before testing to see if it works. See: Instructables  and Re-surfacing So They Work Again 
- For extreme cases: When nothing else works, you can try a heat and boil method (See: instructables.com ). Once you’re finished you’ll want to make sure that the CD is not warped at all since it could damage your disk drive. I would try this if desperate just to make a copy and use that backup copy instead of the original.
Your goal is to buff or work out the damaged areas, not fill them. To do that you’ll likely need to work in circular motions, but doing so may create new light scratches so you’ll need to repair those as you go. If you can manage to work them out by simply moving in an inner to outer motion (from the center of the disc to the outer edge, no circular motion), that would be ideal but it’s not feasible in all situations.