One of the best Italian restaurants that I've eaten at in NYC was a place on Mulberry Street (and I hate Mulberry St, it's way too touristy) called Il Cortile. It is a highly rated restaurant and comprised mostly of local New Yorkers, as not many tourists frequent the place.

BEST in NYC- Greenwich Village, Mario Batali's place called Babbo. Hands down the most authentic Italian food I've ever had in the city.