Final answer:
A writ of certiorari is a request to the Supreme Court to review a lower court's decision; it requires the agreement of four Justices to be granted. The Supreme Court usually grants certiorari when there are significant legal questions or discrepancies in law application, and it heavily relies on written briefs and the lower courts' records to make its decisions.
Step-by-step explanation:
A writ of certiorari is a procedural tool through which a higher court, usually the Supreme Court, may review the decisions of lower courts. The Supreme Court predominantly exercises this power of judicial review when there are substantial questions of law, especially those concerning the Constitution or when there is a discrepancy in the application of the law across various jurisdictions. When a petitioner requests a writ of certiorari, they are essentially asking the Supreme Court to examine the record from a lower court and to schedule the case for argument.
For the Supreme Court to grant a writ of certiorari, at least four Justices must agree to hear the case, which is the so-called Rule of Four. The Court's discretion to choose cases is guided by certain considerations, such as conflicts between lower courts on the same matter or urgent matters that require prompt legal resolution. Once the Supreme Court agrees to hear a case, both parties submit detailed briefs, which the Justices study along with the record of the lower court proceedings and any amicus curiae briefs received, to prepare for oral arguments.