Damages are the official amount of money awarded to the winner in a law suit. If you're hurt in a car accident that's another driver's fault, you may end up collecting damages.
The legal term damages usually refers to money that's owed or received to compensate someone for a loss or an injury. If a business owner's negligence — not clearing ice from a sidewalk, for example — causes you to be injured, you can sue her for damages to pay for your broken arm and the days of work you had to miss. Sometimes a court awards "punitive damages" as well, which are specifically intended to punish the defendant.