When something is "damaged", it means that it's broken, messed up, or isn't working as it is supposed to. "Damage" is the noun form.

Permanent damage that occurs in a certain part of your body is called "__ damage":

The doctor suspects that there may be brain damage.

It contains a chemical that is known to cause nerve damage and irregular heartbeat.

"Liver damage" is a common health problem for people who drink too much for many years.