Carthage is a village in the town of Wilna in Jefferson County, New York, United States. The population was 3,236 at the 2020 census. The village is named after the historic city of Carthage.

The administrative divisions of New York are the various units of government that provide local services in the American state of New York. The state is divided into boroughs, counties, cities, towns, and villages. They are municipal corporations, chartered (created) by the New York State Legislature, as under the New York State Constitution the only body that can create governmental units is the state. All of them have their own governments, sometimes with no paid employees, that provide local services. Centers of population that are not incorporated and have no government or local services are designated hamlets. Whether a municipality is defined as a borough, city, town, or village is determined not by population or land area, but rather on the form of government selected by the residents and approved by the New York State Legislature. Each type of local government is granted specific home rule powers by the New York State Constitution. There are still occasional changes as a village becomes a city, or a village dissolves, each of which requires legislative action. New York also has various corporate entities that provide local services and have their own administrative structures (governments), such as school and fire districts. These are not found in all counties.