When you damage something you hurt it in a way that makes it less valuable or useful. Say you throw a baseball and it flies over your friend's head and cracks your dad's windshield. Congratulations, you've damaged your dad's car.
Damage can also be a noun. That crack in the windshield? It doesn't matter whether you 'fess up or blame your friend or a giant falling acorn. It made the car less valuable and harder to drive, therefore it is considered damage. Another meaning of damage implies the damage done to your wallet when you buy something. In a restaurant, you might hear someone ask "What's the damage?" when the check comes to the table.