New York City has zero villages. It's composed of five boroughs, each coextensive with a county:
In New York State, a "village" is an incorporated municipality within a town (e.g., Scarsdale, Port Jefferson). NYC's charter eliminated towns and villages within its boundaries in 1898 (consolidation). Neighborhoods like Greenwich Village, East Village, or West Village are informal names, not legal entities.
So: 5 boroughs, 59 community districts, hundreds of neighborhoods, 0 villages.