Tell if a CD Is Damaged

Through these series of tests, you will learn how to determine if a disc is damaged, and in some cases possible solutions to fix it.

Steps

Make sure the CD player is working. To do this, try several different CD's in the player. If possible, use new ones that you know aren't scratched.
Play the music CD and listen for parts that skip. If you find some, go on to the next step. If you do not hear any skips, then your CD is not damaged and you may stop here.
Clean the plastic surface with a soft cloth. Move radially from the center to the edge of the CD - never in circles! The reason for not moving in circles is that CD's store data redundantly in circles. If you scratch it from center-out, chances are there's no data loss. But if you scratch it in circles, chances are you'll ruin it.
Play the CD again. If there's still a problem, keep reading.
Flip it over and look for scratches.
Repair light scratches. If you find light scratches, buy a CD Repair Kit and use it. If you find big scratches, go to the next step.
Make a hard decision for big scratches. If you get to this step, either you just throw your CD in the trash or ask at an electronics store for the best CD Repair Kit they have. It may be expensive.

Or, check out this Fix-a-Scratched-CD.
You can buy a CD Repair Kit at any store that sells CD's or almost every supermarket.
There are some CD Cleaning Kits too.
For audiophiles: If you're lucky you might be able to find a copy of the CD available as a FLAC archive online. You may be able to download this on the basis of fair use, but as always, download at your own risk.
A CD not working in one CD player/computer may work fine in another player which is of a different make.
There's a last ditch option for the technologically savvy: Stick the CD into your computer and convert it to an MP3 or other format, and then just listen to it on an MP3 player. Most Audio conversion software programs are better able to use the redundancy features of data on the CD and salvage the data on the disc. In other words, often a CD that won't play in a CD player will convert to MP3 and play just fine in an MP3 player. This happens quite often and is worth a try as a last resort.

If you really can't be bothered to put it back in its case, make sure to keep it label-up.
Don't leave the CD outside of its case!
When using the CD drive of a computer, make sure that the disk has little to no visible damage before putting it in.