The state of New York is a very large state. The area you are referring to is the downstate area or the New York City metropolitan area.
New York City is made up of 5 boroughs (similar to counties): Manhattan, Brooklyn, Queens, the Bronx, and Staten Island. NYC's population is approximately 8 million people. The metropolitan area of New York City consists of Westchester, Fairfield, and Rockland counties to the North, Nassau and Suffolk (Long Island) counties to the East, and Northern NJ. All together, the population of Metro NYC is approximately 18 million people.
Manhattan, which most people refer to as New York City, is made up a many different neighborhoods, which are pretty well-defined on a map.
The Upper East side is known for its "blue blood", aka old money.
The Upper West Side is more liberal and recently has become more family-oriented.
Soho is known for its shopping and art galleries.
50 years ago, Little Italy was a much larger neighborhood. Now, its been pretty much condensed to Mulberry Street because of the growth of Chinatown.
Midtown, is home to Times Square and the Theater District (Broadway).
Greenwich Village, was formerly a very gay neighborhood but has now become more of a tourist attraction and family oriented.
Chelsea has become the gay neighborhood.
The East Village and Hell's Kitchen or Clinton, once the cheapest places to live in Manhattan below 96th Street, are now very trendy.
Harlem is NYC's center of Black-American culture.
Many other neighborhoods to visit, in Manhattan and all the boroughs. Most are accessible by subway-which is very safe.
If you are planning a trip and need any more info, e-mail me. I'll be glad to help.