When you say "New York City," I assume you primarily mean Manhattan (I figure you'd be more specific if interested in the other boroughs) Here's a good, general guide to neighborhoods in Manhattan.
Even though all the neighborhoods are located very close together, they all have a different feel to them. Turning a corner can feel like going into a completely different town. There are also many sections that specialize in certain kinds of people. There are areas dominated by artists. Areas dominated by specific ethnic groups. Gay areas, ect. Different streets also vary widely in how quiet or loud they are. There's something for everyone there, so there's really no "best" area.
Zooming out further though, Manhattan would probably be the "best" borough and is usually the one people think of when they hear "New York City." The other boroughs are cheaper, but more like suburbs than the city. However, you're much more likely to find affordable housing in them if you're on a budget. Despite not being in the heart of the city, the subway system reaches far into them, so you won't have to worry about driving to Manhattan, should you chose to live in one of them.