The formal request is called a Petition for a Writ of Certiorari.
The Court grants certiorari to the petitioner, and issues a writ of certiorari to the lower court, asking for the case files.
Most cases are appealed to the US Supreme Court by a petition for a writ of certiorari, which is a request that the justices accept review of the case and issue a writ of certiorari, or order to the lower courts to send all trial and appellate records to the Supreme Court. Review of an appeal is not a right; the justices grant certiorari at their discretion.
Appellate courts may also issue a writ of error, which is an order to release the trial record of an adjudicated case. This is most often sent from an intermediate appellate court to the court of original jurisdiction.