Manhattan is really surrounded by three rivers: the Hudson River, the East River, and the Harlem River. Except that only the Hudson is actually a river.
The Hudson River flows down the west side of Manhattan. Its west bank is formed hereabouts by New Jersey, with Jersey City opposite the tip of Manhattan and Hoboken (birthplace of Frank Sinatra) to the north of that.
Wikipedia describes the East River as a "salt water tidal estuary". This, apparently, is "a partially enclosed coastal body of brackish water with one or more rivers or streams flowing into it, and with a free connection to the open sea."
The East River's principal tributaries are the Bronx River (which, as its name implies, flows through the Bronx in the north) and the Flushing River (which flows off Long Island, to the south). Flushing is a district of Queens, and owes its name to the Dutch port of Vlissingen. It gives its name in turn to Flushing Meadows and Flushing Bay, as well as the Flushing River – which is historically and more properly known as Flushing Creek.
The East River separates Long Island (to the south–east) from Manhattan (to the north–west). The areas of open sea that it connects with are Long Island Sound (north–east) and Upper New York Bay (New York Harbour – south–west).
The Harlem River separates Manhattan from the Bronx (which is on the mainland). Wikipedia describes it as a "tidal strait".
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