Many posts on this site discuss “Metropolitan New York.” There are two very common ways to draw these boundaries, and your choice generally depends on what’s available in the data. The map of this region below is reproduced from a Census Bureau-published map that can be downloaded as a PDF here.
New York City. The City of New York includes the five boroughs: Manhattan, Queens, Brooklyn, Bronx, and Staten Island. It includes only a subset of the population and economic activity taking part around the City itself.
Sometimes, a strict focus on New York City itself will create a partial picture of the larger environment in which people or organizations operate. Many organizations serving people New York City have workers, suppliers, clients, or constituents that are outside of the formal city limits. If we want to incorporate a view that includes activities taking place around NYC, we might consider looking at an entire CBSA or CSA.
Core-Based Statistical Area (CBSA). This region includes areas with the same shade of green as the City itself. It includes all of Long Island, North New Jersey, the upper parts of the Jersey Shore, and New York’s Westchester, Rockland, Putnam, Orange, and Dutchess counties.
Combined Statistical Area (CSA). The largest conception of this region is the Combined Statistical Area of New York-Newark, NY-NJ-CT-PA. This includes all green-shaded areas on the map: the NY Core-Based Statistical Area, plus additional upstate and Western NJ/Eastern PA counties.
The difference between the latter two is somewhat marginal, amounting to population differences of about 19 million vs. 21 million people. In most circumstances, the difference will probably not substantially alter any conclusions that you draw from the data. I generally choose based on data availability and lean on being explicit about which conception is being used.