Criminal court is where you go to face charges if you have been accused of committing a crime or violating a law.  

Municipal Court has jurisdiction of all cases arising under the ordinances of the city or town, and has jurisdiction concurrently with justices of the peace of precincts in which the city or town is located, of violations of laws of the state committed within the city limits. Most traffic offenses, for example, could be handled by a municipal court judicial officer.

Justice of the peace courts have jurisdiction of the following offenses committed within their respective precincts:  Misdemeanors and criminal offenses punishable by a fine not exceeding two thousand five hundred dollars, or imprisonment in the county jail not to exceed six months, or by both a fine and imprisonment. The amount of restitution or incarceration costs shall not be considered as part of the fine for purposes of determining jurisdiction. Justice of the peace courts also hold hearings in felony cases, but only for the limited purpose of commencing an action and conducting preliminary proceedings prior to transfer of the case to Superior Court.

Superior Court is a general jurisdiction court in which some misdemeanor cases are heard, and all felony charges are handled.