2nd Avenue Deli is a New York Institution. Originally located on 2nd avenue downtown, it was reincarnated on the Upper east side and Koreatown. Both restaurants have identical menus and it is still very much a family business with the kind of staff that know their customers and are happy to have friendly banter with diners or those ordering.
Note that the restaurants are open on Shabbat so for many this would not be sufficiently kosher. The restaurants have hashgachas.
While not a date spot, the upper east side location has a terrific cocktail lounge upstairs, fittingly name the 2nd floor.
The food is not cheap as you would expect for kosher meat in New York City, but the portions are generous and the food is terrific.
Particular recommendations include the burger, pastrami, brisket, matzo ball soup, split pea soup, chopped liver, tongue polonaise and the tuna salad and health salad, which are surprisingly excellent. The pareve desserts are tasty as well and the meal is capped with egg creams. On a cold day or if you just want an authentic Jewish meal, order a soup and sandwich.