Final answer:
The Supreme Court commonly uses appellate jurisdiction to hear cases on appeal, where parties petition for a writ of certiorari. If granted by at least four justices, the case is reviewed; otherwise, the lower court's ruling stands. The Court's decisions are final and set precedents within the federal judicial system.
Step-by-step explanation:
The most common way for the Supreme Court to address cases appealed to it is by utilizing its appellate jurisdiction. Parties from lower court decisions can request the Supreme Court to grant a writ of certiorari, which requires at least four of the nine justices to agree to hear the case, known as the Rule of Four. If the writ is granted, the case is added to the Court's docket for review. Should the Court decide not to grant certiorari, the lower court's decision stands. In rare instances, the Supreme Court exercises original jurisdiction in cases involving states or ambassadors.
Once the Court agrees to hear a case, the justices review the case's record, hold oral arguments, and confer privately to deliberate and vote on a decision. The Chief Justice speaks first in these conferences, followed by the other justices in order of seniority. This decision process ultimately decides the remedy and resolution of the case, and the Supreme Court's decision is final, making it the 'court of last resort'.
The Supreme Court's role in hearing cases on appeal and its power to declare laws and executive actions unconstitutional are foundational to the operation of the federal judicial system, which includes district courts, circuit courts, and the Supreme Court as the final arbiter.