Oh gosh, there are soo many...
Frankie & Johnnie's Steakhouse is in the heart of Times Square. It's pricey but it's worth it. The steaks are incredible. Peter Luger's is another famous steakhouse.
Elaine's is on the Upper East Side, but it's a fun place--you can look it up on Wikipedia. The food isn't amazing--it's known for the people it attracts, not the food. It used to be Frank Sinatra's main hangout, and you might catch a celebrity there.
P.J. Clark's (Clarke's?) has incredible burgers. I would definitely recommend going there.
Papaya King. Yeah, it's basically just hot dogs, but it's super famous (and the hot dogs are great)
J.G. Mellon's is fun.
Lombardi's is known to have some of the best pizza in NY. I don't particularly like it, but lots of people do.
Oceana for seafood--or The Red Eye Grill.
Michael Jordan's in Grand Central is good, though a tad expensive.