This refers to your actual out-of-pocket losses. When you sue for compensatory damages, you're suing for the exact amount of money you lost. You're asking the court to "compensate" you for your losses. If you were supposed to be paid $5,000 on a contract, and you had to sue to recover this money, your compensatory damages would be $5,000.
This refers to a certain amount of money -- usually quite a bit higher than the compensatory damages -- to punish the offender for having injured you or damaged you in the first place. Punitive damages are intended to set an example, to send a message out to the public that the offenses committed by the other side are contrary to public policy. Punitive damages are designed to deter such conduct in the future.
Another term for punitive damages. Exemplary damages serve to make "an example" out of the offending party.
Sometimes, the right to damages is spelled out in a federal or state law. For example, if a police officer violates your civil rights and you sue the police department and win, you may be entitled to "statutory damages" under the Civil Rights Act.
Sometimes referred to as "triple damages." This refers to three times the amount of compensatory damages. Treble damages are a form of punitive damages. They may also be a form of statutory damages if specified in a published statute.
Usually a small amount, such as one dollar or ten dollars, paid by a jury to make a point (but where there is not meaningful loss).