Damages

In law, damages are an award, usually of money. It is to be paid to a person as compensation for loss or injury. The rules for damages can and frequently do vary based on the type of claim which is presented (e.g., breach of contract versus a tort claim) and the jurisdiction. Damages are not the same as court costs which includes the costs to bring a legal case to court. The losing party may also have to pay court costs.
Categories.
In common law countries, damages are usually categorized into compensatory (or actual) damages, and punitive damages. Compensatory damages are further categorized into special damages and general damages. Special damages include economic losses such as loss of earnings, property damage and medical expenses. General damages are noneconomic damages such as pain and suffering and emotional distress. Statutory damages are those required by statute.
Compensatory damages.
Compensatory damages are paid to compensate the claimant for loss, injury, or harm suffered as a result of another's negligence (tort law). Expectation damages are used in contract law. In civil law countries, these are called "ordinary damages" and may be limited to actual loss.
Punitive damages.
Generally, punitive damages, which are also termed "exemplary damages" in the United Kingdom, are not awarded in order to compensate the plaintiff, but in order to reform or deter the defendant. They are also to intended to be an example to others not to do the same thing. Punitive damages are awarded when it is determined that compensatory damages are not enough. Punitive damages are awarded over and above the amount of compensatory damages, such as in the event of malice or intent. Great judicial restraint is expected to be exercised in their use. In the United States punitive damages awards are subject to the limitations imposed by the due process of law clauses of the Fifth and Fourteenth Amendments to the United States Constitution.
Punitive damages in most civil law countries are not awarded. 
Statutory damages.
Statutory damages are an amount stated within the statute rather than calculated based on the degree of harm to the plaintiff. Lawmakers will provide for statutory damages for acts in which it is difficult to determine the value of the harm to the victim. Someone breaking the law can entitle the victim to a statutory award. This is even if no actual injury happened. 
For example, United States Civil Code 18 USC §§2520 provides for statutory damages to victims of various wiretapping offenses.