Answer: Compensatory damages compensate the injured party (plaintiff/claimant) for all of their damages (medical bills, lost wages, pain & suffering, etc).
Punitive damages punish the at-fault party (defendant) for their behavior so as to deter such behavior in the future.
Punitive damages only apply in specific circumstances, usually drunk driving or other conduct that exhibits a blatant disregard for human safety.
Although punitive damages are imposed upon a defendant, they are paid to the plaintiff.
While some liability insurance policies do, most liability insurance policies exclude coverage for punitive damages, meaning the defendant would be personally responsible for paying any punitive damage imposed against them.