The use of the adjective "supreme" suggests that the U.S. Supreme Court is beyond review, and, in fact, it is. The Court has the final say in interpreting federal law. The decisions of the highest court in the land cannot be overturned by any other part of the judiciary. Like the Pope, its decisions can be considered "infallible," even if you think they are wrong.
TL;DR (Too Long; Didn't Read)
The United States Supreme Court hears and decides appeals from federal circuit courts and from state courts involving federal law, as well as constitutional challenges to actions by the legislative and the executive branch.
What Does the Supreme Court Do?
The Supreme Court of the United States is called the highest court in the land because it makes the final decision on questions of federal law, including constitutional issues. It was established in the Constitution as the third branch of the federal government, the judiciary. The nine justices are appointed by the President and confirmed by the Senate. They serve for life.
The Supreme Court hears appeals from the next highest rung of courts in the federal system, the 13 federal circuit courts. It also can hear appeals from the highest state courts in cases involving federal law. Although the bulk of the cases the Court hears are appeals, it also has original jurisdiction. That means that certain cases can be brought directly to the Supreme Court. These include cases in which a state is a party and cases involving ambassadors.
How Does the Supreme Court Decide Which Cases to Hear?
The Supreme Court doesn't have to plow through any appeal that comes its way. In most appeals, the court gets to pick and choose its cases. A party who wishes to appeal a decision to the Supreme Court must file a writ of certiorari. This document is a request for the Court to review a case. The Certiorari Act of 1925 granted the Supreme Court the right to decide whether or not to do so. The Court agrees to hear about 100 to 150 of the some 8,000 cases that it is asked to review each year.
How does the Court make its decisions on which writs to accept? Each certiorari request is sent to each justice, then, led by the chief justice, the court puts together a “discuss list” of potential cases. The decision to grant certiorari requires the vote of at least four of the nine justices. The cases selected often involve important federal questions or issues that have been decided in contradictory ways by different circuit courts.
What is an Oral Argument?
Once the Court has made a decision to hear a particular case on appeal, it notifies the parties. They arrange for the lower-court records and briefs to be sent over and also prepare and file legal documents called briefs setting out their positions. Interested third parties can also submit amicus curiae briefs. As an appellate court, the Supreme Court doesn't hear witnesses or see evidence. They decide the matter on submitted briefs and records, followed by oral argument.
Most of the time, each party gets 30 minutes of oral argument time to set out their positions. Each party tries to persuade the Court that the law should be interpreted in a way that supports its point of view.
What Does the Supreme Court Do When Not in Session?
The term of the Supreme Court is set by law to run from the first Monday in October in one year until the first Monday in October of the next year. About 8,000 new cases are filed each term. During part of each term, the Court hears cases and delivers its opinions on prior cases. These are called "sittings." They alternate with "recesses." During recesses, the Court doesn't hear any new cases but rather considers the pending cases and writes opinions. The sittings and recesses alternate with one another during the term at approximately two-week intervals.
Things change for the Court in May and June. During that time, the Court doesn't hear new cases. It sits only to announce orders and opinions. When July rolls round, the Court recesses until October. During recess, the Justices get to relax a little but not entirely. They keep on considering new petitions for review and other motions and applications. They also work on cases that are on the schedule for argument in the fall.