Italian -- Carmines in Times Square. The servings are enormous - one entree is plenty for 2 people to share.
Dessert or breakfast -- Junior's makes a really great cheesecake. Breakfast is good, but I usually just grabed a coffee and an egg sandwich at a bodega.
Deli sandwich -- Katz's. Life changing pastrami on rye. My sister and I shared one sandwich for lunch and staggered out in a food coma. I would go back to NYC just for another of those sandwiches.
My sis and I spent 5 days in NYC before flying on to Iceland, and those are the 3 places that stick in my memory. Seriously, just follow your nose down the street and you'll find great meals.
I'm going to give a special shout out to Bouchon Bakery in Rockefeller Center because they were so very kind to me. I had a monster nosebleed near their stall, and they got me a chair, a huge pile of napkins, and helped my sister find an urgent care center nearby. (My nose was gushing) They even called the cab. They never said a word about the mess.
(That nostril was cauterized and has never caused problems again. The clothes I was wearing were not salvageable.)