Someone who is from New York City or Long Island.
Yes, Upstate New York is part of the state and technically they are New Yorkers, but when someone says "I'm a New Yorker." people generally think NYC or LI.
Guy 1: I'm a New Yorker.
Guy 2: Oh, you're from the city?
Guy 1: No, I'm from Syracuse.
Guy 2: ...You're not a New Yorker.