According to the Constitution, the SCOTUS has both appellate and original jurisdiction. Original jurisdiction means cases that are heard for the first time in the Supreme Court. These are usually high ranking cases that involve disputes between states or between high ranking officials.
Appellate cases make up the majority of cases heard by the SCOTUS. These are cases that have been heard before by lowers courts throughout the country. Unsatisfied parties with a lower court judgment submit a petition to the SCOTUS, called Writ of Certiorari. The SCOTUS then has all the files from the lower court case sent to it for examination. Four of the nine Justices must accept to hear a case for it to be heard by the SCOTUS. If the Justices accept the case, it is placed on the docket of cases to hear. The petitioner is granted a certain amount of time to write his brief on the case (no more than 50 pages). The opposite party, called the respondent, also has some time to submit its 50 pages brief.
After this initial process, both parties respond to each other’s brief with a shorter brief. The SCOTUS may also grant permissions to groups that are not directly involved in the legal dispute to provide their own file with recommendations and arguments in favor of the side they support. Such brief is called amicus curiae (friend of the Court, in Latin). The SCOTUS hears the oral arguments for both parties sometime between October and April. These proceedings are open to the public and usually start at 10am. Each hearing lasts an hour and each party has 30 minutes to present its argument. The hearing is interrogative (Justices ask questions to each party about their position). Petitioners speak first and respondents follow suit. If petitioners have saved time for a rebuttal, they will be the last to speak.
The Justices meet twice every week to review the cases and vote on a decision. This meeting is called the Justices’ Conference. After the result of the vote is available, any dissenting Justices may write a dissenting opinion on the final decision. If they vote ends up in a tie (because one of the Justices is unavailable), the lower court’s decision stands. In very rare occasions, a majority opinion may be changed if a Justice reverses his/her vote. It is only when opinions are disclosed in public open court hearings that they are official and final.