Explain how cases reach the United States Supreme Court. What types of cases does the Supreme
Court typically review?

There is no absolute right of appeal to the United States Supreme Court. To bring a case before the Supreme Court, a party requests the Court to issue a writ of certiorari. A writ of certiorari is an order issued by the Supreme Court to a lower court requiring the lower court to send it the record of the case for review. Parties can petition the Supreme Court to issue a writ of certiorari, but the Court will not issue a writ unless at least four of the nine justices approve of it. This is called the rule of four. Most petitions for writs are denied. A denial is not a decision on the merits of a case, nor does it indicate agreement with the lower court's opinion. It simply means that the Supreme Court declines to grant the request (petition) for appeal. Furthermore, denial of the writ has no value as a precedent.
Typically, the petitions granted by the Court involve cases that raise important constitutional questions or decisions that conflict with other state or federal court decisions. Similarly, if federal appellate courts are rendering inconsistent opinions on an important issue, the Supreme Court may review a case involving that issue and generate a decision to define the law on the matter.