NYC is made up of five separate areas, called boroughs, and calling them that is a leftover British thing (English metropolitan areas also have boroughs). These areas are Manhattan (the island we call the City and what the rest of the world thinks of when they think “New York City”), the Bronx, Brooklyn, Queens and Staten Island.
Together, they are New York City, but New York state also considers each borough its own separate county. The Bronx is to the north of Manhattan, Brooklyn is to the southeast, Queens is to the west, and Staten Island is south. Brooklyn, Queens, and Long Island are all on one parcel of land, and all of it is geographically Long Island.
“The Big Apple” is old slang for Manhattan coined by jazz musicians back in the 1920s. To get a gig in Manhattan meant you hit the big time, and that’s the Apple you wanna bite of, dig? So, like with any slang, it went from jazz people to their hipster fans and outward to the general populace and it stuck (people have been saying that stuff is “cool” for almost 90 years now. Same sort of thing).