You may be referring to a writ of mandamus or certiorari.
If the appellate court is directing a lower court official to take a particular action, they can issue a writ of mandamus. If they are ordering the lower court to send records for a case on appeal, they issue a writ of certiorari. Your question wasn't clear enough to determine which of the two you're asking about.
For more information, see Related Questions, below.
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Appellate court.
If the US Supreme Court agrees with the lower court ruling, the decision is "affirmed," and becomes legally final (res judicata).
Affirmed
Three major problems with the lower courts: 1) Neglect by bar associations, higher courts, and governmental agencies. 2) The volume and nature of their caseloads. 3) The trial de novo system, the system of new trials, that is, appeals from lower courts to a court of general jurisdiction.
The Justices of the U.S. Court can reverse the decision of a lower court.