You cannot come to New York and not try a deli (short for delicatessen). Carnegie Deli is with Junior's and Second Avenue Deli, one of my favorites. Yes, I did not mention Katz's Deli because I do not like it, think it is overpriced and overrated.
The restaurant opened in 1937 and got its name from being close to the Carnegie Hall. It is still operated by the family of the founder: the Parkers. It is today one of the most visited restaurants in New York City, although the expansion of business has not been that successful, seeing most of the branches in California or Connecticut close. Today, there is a Carnegie Deli in the Mirage Hotel in Las Vegas.