A writ of certiorari is an order from a higher court to a lower court to provide the higher court with the records from a particular case so that the higher court can review the decision made by the lower court. "Certiorari" is a Latin word meaning to be more fully informed. The Supeme Court commonly uses writs of certiorari to chooses which cases it will hear.
Sometimes, if there is dispute about a particular court case, then an appeal may be made to a higher court to re-investigate the matter. This can happen if parties involved in the case believe a mistake has been made, or if people in the legal system believe that another look may be necessary. Such cases often require interpretations of the law. The Supreme Court is the highest court in the system, and therefore, decisions it makes following writs of certiorari have the most authority.