What is now called the West Village is the original Greenwich Village. Greenwich Village was originally a community on the shore of the Hudson River; it was reached from New York across the Lispenard meadows by way of the Road to Greenwich, which is now Greewich Street. The major landowner in the area was Sir Peter Warren, whose daughter married extremely well (she became the Countess of Abingdon), and who celebrated his family's connection to the nobility by erecting a monument to hs son-in-law at what is now Abingdon Square, and a pleasure-drive and walkway that led to it, which is now Greenwich Avenue. When Washington Square was built, it was also considered Greenwich Village, but the idea of a "West Village" only came around by analogy to the "East Village"; which had always been considered part of the Lower East Side up until the 1960s. Because the Lower East Side was considered slummy, those artsy new residents who arrived in the 1950s and who preferred to be associated with nearby Greenwich Village invented the new name "East Village". As this took hold, the idea grew that the center of Greenwich Village was Washington Square, and what had originally been Greenwich Village in its entirety started to be called the "West Village."
Brooklyn is not a neighborhood, but is instead a borough that is three times the area of Manhattan with a million more inhabitants and many more neighborhoods. You therefore need to decide what part of Brooklyn you want to explore.
