The City refers to
Manhattan OR the five boroughs. If you live in the boroughs, then 'The City' is definately Manhattan. Outside of the boroughs, it can mean the boroughs as a whole or just Manhattan.
Remember: New York City isn't just
Manhatten, it's the whole five boroughs (Brooklyn, The Bronx, Queens,
Staten Island and Manhattan).
Brooklyn kid 1: lets go
the the city tonight.
Brooklyn kid 2: Yea, I saw something I wanted to get on 40-duce.
Out of
stater 1: Lets visit the city!
Out of stater 2: I don't know, its a big place, and I hear it's expensive.