There are tons of options for pizza in NYC, but I'm particularly blessed living in the East Village where there are two incredible spots a stone's throw from the corner of East 12th St and 1st Avenue. In no particular order, there is Motorino on E 12th St and 1st Ave and Luzzo's on 1st Ave between E 12th and E 13th Sts.
Both Motorino and Luzzo's produce delicious thin crust Neapolitan-style pizza. Motorino cooks their pizzas in a brick oven, while Luzzo's has a 120+ year-old coal-burning oven. Coal-burning ovens are rare in NYC as no new ovens are allowed, they must be inherited!
Motorino's strength: a focus on fresh ingredients. Their Soppressata pizza is outstanding, as well as their seasonal specials. Their restaurant is small, charming, and causal and they usually have a line out the door.
Luzzo's strength: Their coal-burning oven does magical things to their crust. Their Salsiccia E Broccoli is phenomenal. Luzzo's is causal and charming as well, but it is a slightly larger restaurant and thus seating is easier to get. They also allow BYOB with a $15 corkage fee.
You really can't go wrong with either. And if one has a long wait, go to the other for amazing pizza. They both have locations in Brooklyn and beyond (see their websites).