New York City aka the "Big Apple" is the biggest city of the world in 1940, with about 7.5 million people, surpassing former record holder London.
Only as late as in 1898, the core city of Manhattan was joined by the other boroughs of Brooklyn, the Bronx, Queens, and Richmond aka Staten Island. Older New Yorkers can still remember the time before.
The pride of the city - until an even higher skyscraper will be built
The city is unique for its skyscrapers, especially the tallest one, the Empire State Building which was finished quite recently, in 1931. After that, the Great Depression prevented people from building higher and higher skyscrapers to one-up each other. Only Adolf Nazi thinks about building a skyscraper in Hamburg which is planned to become even taller - but doesn't consider that Hamburg is built on soft soil, as opposed to Manhattan's bedrock.
Not only are the theatre shows, jazz bars and musicals shown at Broadway world-famous (although disparaged by the Nazis), the city also makes a great place for authors, like F. Scott Fitzgerald. Furthermore, it's also a capital for the media, like the New York Times and the New Yorker. Only the moviemakers prefer to stay in Hollywood.
Besides for economy and culture, the city also is home of famous universities, like Columbia U. Where unbeknowst to Nazi Germany, Italian emigre Enrico Fermi is working on secret nuclear projects.
The city has not one but two airports, Newark Metropolitan Airport (since 1928) - first one in the world where commercial flights can take off - and New York Municipal Airport-LaGuardia Field (since 1939), and some people already think about building a third one.
On the other hand, there are also drawbacks: The city is a hotspot of organized crime, although Thomas E. Dewey did a lot about that during his time as a prosecutor and District Attorney, having brought gangster bosses like "Lucky" Luciano to prison, and murder rates have been about halved since 1931.
Nazi propaganda paints the city as being controlled by Jews, esp. the Wall Street, and being practically a "colored" city. A huge exaggeration: While it has a multi-cultural population (besides about two million Jews and many blacks, there are also many Italians, Irish, Poles, Chinese, and in fact dozens of other ethnics - Germans too who still celebrate Steuben Parade every year), only 8% are actually non-white. Still, the city's truly the opposite of everything the Nazis want.
Its current mayor (since 1934) is Fiorello Enrico "Henry" LaGuardia. He has city planner Robert Moses making plans for a big redesign of the city, involving tearing down whole quarters to make place for city highways, to the chagrin of people like Jane Butzner who love the grown structures of their quarters.