Cases come to the Supreme court in two ways:
- Under original jurisdiction, which currently includes only disputes between the states.
- Under appellate jurisdiction, where the case (usually) has exhausted all appeals in the federal or (sometimes) state court system.
Basically, a case can go to the Supreme Court through appeals. Another way a case can make it to the Supreme Court is if it involves changes to the federal law.
Cases are not directly filed with the US Supreme Court. They must begin in the Federal Circuit in one of the US District Courts or in a state trial court.
If a verdict unsatisfactory to one side in the case is rendered, the next step is to appeal that case to the US Federal Court of Appeals for the Federal District in which the District Court was located, or to the court of appeals for the state in which the case was originally tried.
If the verdict is still unsatisfactory, it may be submitted to the US Supreme Court (unless the case originated in the state court system, in which case it must be petitioned to the state supreme court first) which will then decide on whether it wishes to hear the case, or not.
If not - they will remand it back to the Court of Appeals for final jurisdiction. If they DO accept it, they will hear arguments from the attorneys for both sides, and then each Justice will render their own opinion on the case. The majority of opinions decides the case.