In some states, the name of an upper court where civil lawsuits above the Small Claims Court limit are filed, similar to the Circuit Court or District Court in other states.
Merriam-Webster Online Dictionary
district court (noun) a trial court that has jurisdiction over certain cases within a specific judicial district
District Court (Wikipedia)
District courts are a category of courts which exists in several nations, some call them "small case court" usually as the lowest level of the hierarchy. These include:
Allegheny General District Court, Covington, Virginia.