Cases that deal with the constitutionality of a law;
Cases involving the laws and treaties of the U.S.;
Cases involving ambassadors and public ministers;
Disputes between two or more states;
Habeas corpus issues.
Most criminal cases, probate (involving wills and estates)
Most contract cases, tort cases (personal injuries), family law (marriages, divorces, adoptions), etc.
State courts are the final arbiters of state laws and constitutions. Their interpretation of federal law or the U.S. Constitution may be appealed to the U.S. Supreme Court. The Supreme Court may choose to hear or not to hear such cases.