I like my crust puffy, chewy, and pliant. A great pizza crust should be like a great football defense. It should bend but never break. The superior pizza crust is neither cracker-thin nor thick as bread. It should have a veneer of crispness and be softer and more tender on the inside. A great pizza crust should have browned and blackened char spots. They lend a needed bit of smoky flavor. I love a pizza crust with a few of those raised blisters. They lighten the overall effect of eating pizza. The interior of the crust should have the hole structure of well-made and well-baked bread.
Pizza is as simple a great food as there is. But a pizza crust becomes great not just because of the quality of ingredients but also because of the skill of the pizza-maker. It's all in the hands.
I love the clean milky taste of fresh cow's milk mozzarella (fior di latte in Italian) on my pizza. It's easy to tell if a pizzeria uses fresh mozzarella: It's white. Aged mozzarella, used by the great majority of pizza makers in this country, is that color I call "pizza yellow." Because it's aged, it has a certain tanginess. More and more pizzerias in NYC and across the country are making their own whole-milk mozzarella fresh every day.
Most good by-the-slice places use a commercial aged mozzarella made by Grande, a large cheese purveyor in Wisconsin. Mozzarella made from the milk of water buffaloes is called mozzarella di bufala. It's tangy, very creamy (it's actually wet and quite oily when it melts), slightly tart, and unbelievably delicious. It's also quite expensive (even in Italy, where most of it is made), and you'll rarely see it on a sliceria slice — except sometimes at Di Fara in Brooklyn.
The best pizza sauces are made with uncooked canned tomatoes, from either California or Italy, that have been strained and seasoned with salt and maybe some oregano. Pizza sauce should not be slow-cooked. It should not taste like pasta sauce or marinara sauce. Some people think you have to use San Marzano tomatoes, grown in that town in Italy, that have been designated DOP (Denominazione di Origine Protetta). They are very expensive and I have found wonderful pizza sauces made with other Italian tomatoes or even high-quality California tomatoes. Some people doctor their sauce with sugar, giving it an excessive sweetness that I don't care for. If the tomatoes used are good enough, they will be plenty sweet on their own. Fresh tomatoes are generally not used to make the sauce for pizza, though fresh cherry tomatoes are used to make the pie the Neapolitans call al fileto.
[Thanks, Ed! I, Adam, will jump in here now and say that these descriptions apply to great pizza in general, not just NYC slices, but they are great guidelines on what to look for when you're looking for great slices or pies in NYC or anywhere. —AK]