Historically, certiorari was a prerogative writ used to direct a lower court or tribunal to certify for review the "record" in the case.
In the U.S., certiorari is the writ that an appellate court issues to a lower court in order to review its judgment for legal error, where no appeal is available as a matter of right. Since most cases cannot be appealed to the U.S. Supreme Court, for example, a party who wants that court to review a decision of a federal or state court files a "petition for a writ of certiorari" in the Supreme Court. If the court grants the petition, the case is scheduled for briefing and argument. That does not necessarily mean the Supreme Court has found anything wrong with the decision, merely that it wants to look at it for some reason. Four of the nine justices must vote to grant a writ of certiorari. The great majority of cases brought to the Supreme Court are denied certiorari, because the Supreme Court is generally careful to choose only cases in which it has jurisdiction and which it considers sufficiently important to merit the use of its limited resources. Certiorari is sometimes informally referred to as cert, and cases warranting the Supreme Court's attention as certworthy. One situation where the Supreme Court sometimes grants certiorari is when the federal appeals courtss in two (or more) federal judicial circuitss have ruled different ways in similar situations, and the Supreme Court wants to resolve that "circuit split" about how the law is supposed to apply to that kind of situation.
Some U.S. state court systems use the same terminology, but in others, writ of review or leave to appeal is used in place of writ of certiorari as the name for discretionary review of a lower court's judgment.
In the administrative law context, the common-law writ of certiorari was historically used by lower courts in the U.S. for judicial review of decisions made by an administrative agency after an adversarial hearing. Some states have retained this use of the writ of certiorari, while others have replaced it with statutory procedures. In the federal courts, this use of certiorari has been abolished, and replaced by a civil action under the Administrative Procedure Act in a United States District Court, or in some circumstances, a petition for review in a United States Court of Appeals.