On the state level, the court that has a case is known as the court of?

The federal district courts hear cases that arise under federal law or the U. S. Constitution. The second levels are the appellate courts, which hear appeals from the trial courts. Both the state and federal systems have a Supreme Court, to serve as the "court of last resort."

In the United States, a state supreme court (known by other names in some states) is the ultimate judicial tribunal in the court system of a particular state (i. e., that state's court of last resort) ... Generally, the state supreme court, like most appellate tribunals, is exclusively for hearing appeals of legal issues.

the levels from lowest to highest are District Court they hear your criminal/civil cases. Second is Appellate courts where they mostly hear appeals. The Supreme Court is the highest of them all and hears Petitions to deem unconstitutional, handles cases regarding foreign ambassadors and appeals but it is hard to get a appeal heard in the supreme court.