What kind of jurisdiction does the US Supreme Court exercise?

The US Supreme Court exercises original jurisdiction and appellate jurisdiction.

The Supreme Court is vested with the judicial power of the United States. Judicial power is in the form of original jurisdiction (trials) and appellate jurisdiction (appeals and reviews of trial decisions) The Supreme Court has original jurisdiction in cases involving ambassadors other public ministers and consuls and those in which a state shall be a party and appellate jurisdiction in all other cases. See Article III, Section 2.

Article III, Section 2

"The judicial power shall extend to all cases, in law and equity, arising under this Constitution, the laws of the United States, and treaties made, or which shall be made, under their authority;--to all cases affecting ambassadors, other public ministers and consuls;--to all cases of admiralty and maritime jurisdiction;--to controversies to which the United States shall be a party;--to controversies between two or more states;--between a state and citizens of another state;--between citizens of different states;--between citizens of the same state claiming lands under grants of different states, and between a state, or the citizens thereof, and foreign states, citizens or subjects.

The Court currently only hears cases involving disputes between the States under original jurisdiction; cases involving ambassadors and other foreign officials are first heard in US District Court.

The Eleventh Amendment revoked the Court's right to hear cases between a state and citizens of another state due to problems caused by the Supreme Court's early decision in disputed land grants.

11th Amendment:

"The Judicial power of the United States shall not be construed to extend to any suit in law or equity, commenced or prosecuted against one of the United States by Citizens of another State, or by Citizens or Subjects of any Foreign State."

Federal Question Jurisdiction

The Supreme Court may also hear certain cases on appeal from State courts, if they involve a preserved question of US constitutional or federal law, or issues arising under US treaties. This authority is called "federal question jurisdiction."