Very few cases are overturned on appeal. Above the Appellate Court is one State Supreme Court. Although State Supreme Court decisions are final, on rare occasions, a case can be appealed to the Federal Supreme Court. District Courts are the lower courts and the Trial Courts in the Federal court system. There is at least one district in every state.
Many states have more than one district based upon population.
The district's court of appeals is the Circuit Court. Circuit Courts are numbered. Each Circuit Court serves as the court of appeals over several districts, e.g. the Third Circuit includes New Jersey, New York and Pennsylvania District Courts. Circuit Courts are usually referred to by their number, e.g. First Circuit, Second Circuit, etc. There is only one court above all the Circuit Courts and that is the U.S. Supreme Court.