Boroughs & Neighborhoods

New York City as a whole is much more complicated than your average city. Usually, saying "New York City" is referring to either the five boroughs together or to the borough of Manhattan specifically. The other four boroughs are Brooklyn, Queens, the Bronx, and Staten Island. Each of these five boroughs is broken up into even more individual neighborhoods, and each location has its own personality.

 via NY1 Residential

The image below is a pretty great example of all of the unique neighborhoods in Manhattan. Manhattan is the more touristy part of New York city, and when a friend says they visited NYC,  Manhattan is usually what they mean. Maps tend to differ depending on the source; many of these neighborhoods' borders seem to fluxuate depending on who you talk to. The most common change up I see is the placement of Greenwich Village vs the West Village; on some maps I have even seen them switch places entirely. Also, some maps are far more specific that others. The neighborhoods you will most often hear are the more general areas that people most frequently refer to.

Via http://www.nycinsiderguide.com

The Wikipedia article on NYC Boroughs is very informative.
This website will tell you much more about the unique neighborhoods within New York City.