Question: When a petition is filed before the Supreme Court on a pending case, what is the petition called?
Someone who wants the Supreme Court to review a decision of a lower court files a "petition for a writ of certiorari" asking the Supreme Court to review the ruling of a lower court for errors in law.
The Supreme Court can grant a writ of certiorari, commonly referred to as a "cert," with the vote of four of the nine justices. Most cases brought to the Supreme Court are denied certiorari (about 7500 petitions are presented each year; just 80 to 150 are granted). When the Supreme Court grants a cert, they are not making a decision on the case, they are just saying that the case is important enough and there are enough questions to hear it before the Supreme Court.
For more on Writs of Certiorari and the US Supreme Court, visit the Certiorari page on Wikipedia.