When New Yorkers mention "The Village", they mean Greenwich Village. Located to the north of SoHo/TriBeca and to the west of East Village, Greenwich Village did indeed begin as a country village. City people came to live here to escape a yellow fever epidemic which at that time was gripping the Lower Manhattan part of New York City in 1822.
St Luke's Place, Greenwich Village Source: https://commons.wikimedia.org/wiki/File:19-21_St._Lukes_Place.jpg Author: Djafrican
Greenwich Village, New York City Source: https://commons.wikimedia.org/wiki/File:Greenwich_Village_2005.jpg Author: Seth Werkheiser
Over time, Greenwich Village has evolved into a bohemian haven with a lively gay scene. It is home to many famous artists and writers, and as a result, it has become an expensive place to live. The Washington Square Park and New York University are located here. Many of the apartments in this area are living quarters for the students of the university.
The area that today forms Greenwich Village is bordered by Broadway to the east, West Village to the west, Houston Street to the south and 14th Street to the north. Historically, the southern border of Greenwich Village was at Spring Street. However, with continuous development, SoHo has since encroached into areas formerly regarded as part of Greenwich Village. Adjacent to Greenwich Village is East Village and NoHo to the east, SoHo to the south, and Chelsea to the north.
Street view of Greenwich Village Source: https://commons.wikimedia.org/wiki/File:GreenwichVillage.jpg Author: GK tramrunner229