One of the more common roofing materials are asphalt shingles. Usually they come in the form of tiles that are different shades of gray. While this is a fairly hardy material, there are still ways in which they can get damaged.
With age, the material covering the surface of the tiles could wear off. This damage is not repairable, and the only solution is to replace the tiles.
The tiles could also be damaged by moss or algae growing on them. If you notice this, you should remove it as soon as possibly, and also take steps to prevent future growth. You want to avoid having too much shade over your roof, so it is best to trim back branches that are covering the roof. You should also keep leaves and branches off of the roof, since this will let water collect and encourage growth.
If algae starts growing on your roof, this is not as harmful, but it is very unsightly. Moss and algae can be removed using a 50/50 solution of bleach and water.