The bottom rung of this graphic represents local courts that go by a variety of names - district, county, magistrate, etc. These courts generally hear minor cases and arraignments.
The next rung represents specialized courts dealing with family issues, juveniles,landlord-tenant disputes, etc.
The next level involves state superior courts, where felony trials are heard. Of all trials held in the U.S. each year, the vast majority are heard in state superior courts.
At the top of the state court system are state supreme courts, where appeals of verdicts rendered in state superior courts are heard.