Ones concerning the constitution, federal law, and 2 or more states, mostly
The types of cases the Supreme Court sees involve constitutional issues or federal laws. The Supreme Court gets approximately 7,000 requests to hear cases each and every year.
In most cases, supreme courts are final appellate courts.
State vs. State, or Cases against the U.S.
Having appellate jurisdiction means that the Supreme Court hears cases that have been in trial before. A majority of cases that the Supreme Court hear are either controversial, or some kind of trial error took place in a prior court.
The Supreme Court hears three kinds of cases. Cases appealed from lower federal courts account for two-thirds of the cases they hear. They also hear cases appealed from state's supreme courts, and sometimes hear cases that have not been previously heard by a lower court, such as between one state's government and another.
The Supreme Court always has the last say for cases that fall under its jurisdiction. The Court no longer has mandatory jurisdiction and may exercise full discretion over which cases it hears. The decision of the Court is final, unless modified by the Court itself or by constitutional amendment.
The Supreme Court ruled that the death penalty could only be used in cases involving murder.
The Supreme Court decides cases that are appealed by a lower court; a lower court has made a decision and one of the parties feels strongly enough that the decision was wrong that they make an appeal to the Supreme Court. The Supreme Court reviews the cases and determines which ones they will hear, they have the ability to decline to review a case. The Supreme Court doesn't hear only appeals, there are situations where it is the court of original jurisdiction. In situations where there is a disagreement between states, the Supreme Court has the authority to decide.
The highest court is the Supreme Court, but not all cases can be appealed to the Supreme Court; it depends what kind of legal issues are involved. Otherwise, the case can be appealed to a Federal Appeal Court. If you can afford the legal fees, of course.
They are called supreme court justice
The Supreme Court has original jurisdiction in cases involving ambassadors, other public ministers, and those in which a state is a party. This means that these cases can be brought directly to the Supreme Court without going through lower courts first. Such original jurisdiction is outlined in Article III of the U.S. Constitution. Other cases typically fall under the Court's appellate jurisdiction.
There is no telling which case or what kind of a case the Supreme Court will hear. If a case is simple, it never will get to the Supreme Court. Cases that reach the Supreme Court have gone through one or more appeals processes. Sometimes a appeal reaches the Supreme Court when a federal court of appeals has made a ruling different from another federal court of appeals. In that case, the supreme court is asked to certify an issue. That is a fancy term meaning to play referee. The Supreme Court certifies an issue when it takes up an issue where district courts of appeal have made different rulings concerning the application of the same law. (Sometimes the Supreme Court refuses to take up the issue. In that case it simply states, "Cert. Denied.") Normally, all cases that reach the Supreme Court have come from the Federal Courts of Appeal or the Highest State Court. However, the Supreme Court reserves the right to sit as a court of original jurisdiction. The last time the Supreme Court granted a writ of Habeas Corpus was 1924. It retains that right. I doubt if any member on the Supreme Court has any idea under what conditions that would happen. Still, it retains that right.