If the first appeal was in state court, then the second appeal would be made to the state supreme court. If the first appeal was in federal court, then the second appeal would be made to the United States Supreme Court.
With very limited exceptions, state and the federal supreme courts are not required to take a case. Requests for the highest courts to review an appeal are called petitions for certiorari or petitions for review. Such petitions should be well written to increase the chances of the higher court hearing the appeal.