I have heard many a tourist refer to my neighborhood as a suburb. This not anything that a New Yorker would say. Here is how we break things down in NYC.
First, there is the City as a whole. It is broken down into five BOROUGHS. This is also often spelled boro. Manhattan is but one of the five boros. The others are Brooklyn, Queens, The Bronx and Staten Island. The Bronx is the only boro on the mainland. Staten Island and Manhattan are islands and Brooklyn and Queens are the westernmost two counties of Long Island [the others being Nassau and Suffolk]. In fact, Marble Hill is on the mainland of the Bronx - but in a geographical quirk is part of the boro of Manhattan.
The Boroughs are broken up into NEIGHBORHOODS. We are quite proud of our neighborhoods, and there are lots of differences between them. I live in Park Slope, my neighborhood is surrounded by Sunset Park, Gowanus, Windsor Terrace and Prospect Heights. When a NYer asks me where I live, I would normally say Park Slope, not Brooklyn. When I go to a different neighborhood, like Bushwick, I often feel like a tourist - even in my own boro.
Although many of our neighborhoods in Brooklyn, Queens, Staten Island and the Bronx are green and leafy, and CIty Island in the Bronx just feels like a New England fishing village - NOBODY ever calls them suburbs. SUBURBS are places in Westchester County [New Rochelle, White Plains, Mamaroneck], Long Island [Great Neck, Rockville Centre and Dix Hills], New Jersey [Paramus, Saddle River and Metuchen] and Connecticut [ Stamford, Darien and Cos Cob].
Boroughs, neighborhoods and suburbs. Got it?