(covers information from several alternate timelines)
New York City (also known as New York and the Big Apple) was a city on the Atlantic coast of the North American continent. This city was in New York. Brooklyn and Manhattan were areas within the city. Enforcement of local city ordinances was handled by the New York City Police Department. Major landmarks in the city included the Chrysler Building, the Empire State Building, the Statue of Liberty, the Brooklyn Bridge, the former World Trade Center, and Times Square.
1950s New York, specifically Harlem, was also the home of Benny Russell, the identity Benjamin Sisko assumed in visions from both the Prophets and the Pah-wraiths. It was also the home of the publishing house for Incredible Tales of Scientific Wonder. (DS9: "Far Beyond the Stars", "Shadows and Symbols")
In the 1960s, advanced aliens sent genetically-modified Humans back to Earth to prevent it from destroying itself by carrying out top secret missions using advanced alien technology. They were based in Apartment 12B, somewhere in New York City. (TOS: "Assignment: Earth")
In the year 1944 of an alternate timeline, New York was occupied by Nazi German forces with the aid of 29th century aliens known as Na'kuhl, and received a visit from Adolf Hitler. The leader of the Na'kuhl, Vosk, supervised the secret construction of a temporal conduit in the city. This timeline was corrected by the crew of the Enterprise when they destroyed the conduit. (ENT: "Storm Front", "Storm Front, Part II")
One of New York's most famous streets is Wall Street, former site of the New York Stock Exchange. The Ferengi make annual pilgrimages there, as it was the engine of Earth's old currency based economic system. (VOY: "11:59")
Among the sports teams that represented New York during the 20th century were the Yankees and the Giants (the latter of whom would move to San Francisco). (DS9: "For the Cause", "Far Beyond the Stars")
In Dangerous Ground, Doctor John Rawley, while helping a woman with abdominal pains, when faced with the situation chose to forget what he didn't have and concentrated on what he did have. He believed that the woman was giving birth, until proven otherwise. The doctor once visited a hospital delivery room in the city. According to him, the room was "...beautifully equipped."
Later in the novel, the doctor reflected on his life in Fairbridge feeling that a major disadvantage to living in a small country village was the lack of privacy. He thought, I reflected that in my small New York apartment, with neighbors a few feet away on all sides, I have had much more essential privacy than I ever could expect in Fairbridge. There was much to be said for a metropolitan existence where your life is strictly your own business because no else cares about it. (TNG: "The Big Goodbye")