• Unique New York: Sammy's Romanian and Chumley's (still basically a speakeasy).
• Landmark Taverns: McSorley's Old Ale House, White Horse Tavern, and Pete's Tavern.
• Best Steakhouse: Peter Luger Steakhouse (yes Brooklyn, but just over the Williamsburg bridge from Manhattan). Not cheap, but a no-brainer, as Cutter pointed out.
• World Famous Pizzerias: Lombardi's Pizza, Grimaldi's Pizzeria, and John's Pizzeria.
• Less-famous Great Pizza: Ben's Pizza.
• Confusing "Famous" Pizzerias: The countless "famous original Ray's". Don't bother.
• Delis: Carnegie Deli and Stage Deli.
• Less-famous Delis: Katz's Delicatessen, Barney Greengrass (The Sturgeon King!), and Eisenberg's Sandwich Shop.
• Authentic (Brooklyn) Deli: Jay & Lloyd's Kosher Deli. A schlep.
• Bagels: Ess-a-Bagel, Inc. and H&H Bagels (as seen on Seinfeld).
• Burgers: Shake Shack, P. J. Clarke's, and Michael Jordan's (a little more pricey/touristy, but excellent burgers).
• Hot Dogs: Papaya King, Gray's Papaya, and (at your own risk*) any stand with a Sabrett umbrella.
• Soul Food: Sylvia's (Harlem is tourist friendly).
• Non-food stuff: Strand (books) and Bleecker Bob's (music).

The number-one rule for an authentic NYC dining experience is to avoid Times Square. No natives eat there, and there's probably a Planet Hollywood and a Hard Rock Cafe near where you came from anyway.

* For the truly daring, follow the advice of an old boss of mine and seek out the vendor with the dirtiest fingernails.