My understanding of them are this:
Damage: If something is damaged, it has been broken or is faulty
Danger: There is a chance of something bad happening
Risk: There is a possibility of something bad happening
Peril: Similar to danger, but used when already in a dangerous situation
The definitions are:
Damage: Physical harm that impairs the value, usefulness, or normal function of something.
Danger: The possibility of suffering harm or injury.
Risk: A situation involving exposure to danger.
Peril: Expose to danger.