What are two types of cases that the Federal and Supreme Courts may hear?

Civil and Criminal. The US Supreme Court only has appellate jurisdiction over cases heard in federal District and Circuit Courts.

Or you may mean cases of original jurisdiction (trial) and appellate jurisdiction (appeals).

The only cases the US Supreme Court still has exclusive, mandatory original jurisdiction over are disputes between the states. The Court also has the option of hearing cases involving ambassadors under original jurisdiction, but is more likely to remand the case to US District Court for trial.

[Note: The educational packets that claim the Supreme Court exercises original (trial) jurisdiction over cases involving ambassadors and foreign dignitaries is partially incorrect. Pursuant to 28 USC § 1251(b)(1) (federal law, 1978*).] The US Supreme Court has original, but not exclusive, jurisdiction over such cases:

(a) The Supreme Court shall have original and exclusive

jurisdiction of all controversies between two or more States.

(b) The Supreme Court shall have original but not exclusive

jurisdiction of:

(1) All actions or proceedings to which ambassadors, other

public ministers, consuls, or vice consuls of foreign states are

parties;

(2) All controversies between the United States and a State;

(3) All actions or proceedings by a State against the citizens

of another State or against aliens.

The US District Courts currently exercise original jurisdiction over cases involving ambassadors.

*Amendment by Pub. L. 95-393 effective at the end of the ninety-day period beginning on Sept. 30, 1978, see section 9 of Pub. L.95-393, set out as an Effective Date note under section 254a of Title 22, Foreign Relations and Intercourse.

For more information, see Related Questions, below.