What is the difference between a city, a town, and a village?

There is no national definition, and the distinctions are set by state law, which is why they vary so much. In most states, city, town, village, and borough are types of municipal incorporation with different governmental structures, powers, and formation requirements, and the label often reflects the form of government rather than size. A city typically has a mayor and council or a council-manager structure and broader authority; a town or village may have a more limited government and depend on the county for some services. The same population could be a city in one state, a town in another, and unincorporated in a third.

The counterexamples make the point. Hempstead, New York, is a town with a population larger than most American cities. Several incorporated cities have populations in the double digits. New England towns are the primary unit of local government and often govern through open town meetings where residents vote directly. In some states, notably Virginia, cities are independent of counties entirely, while everywhere else cities sit inside counties. Many populous places, including large stretches of suburban America, are unincorporated and governed directly by the county with no municipality at all.

Internationally the picture differs again. In the United Kingdom, city is a status granted by the monarch and has nothing to do with population, so some cities are small and some large towns are not cities. Globally, statistical agencies use their own thresholds for urban classification, which is why international comparisons of urbanization rates are less precise than they appear. For most practical purposes in the United States, the question that matters is not what the place is called but which government has jurisdiction over it, which determines who provides police, zoning, and services and where you vote.