Gosh, where do I start?? Seems like people already brought up a lot of good ideas. But here's some ideas from a local
Museum of Natural History and the Metropolitan Museum of Art...definite must sees. Even if you don't think you are into that stuff. Definitely check out the dinosaur skeletons at the Natural History Museum. Natural History is at Central Park West and 81st Street, the Met is on 5th Avenue at 86th Street. They are basically both on opposite sides of Central Park from one another, so you can take the subway to one, hail a cab which will whisk you across Central Park on one of the trans-park roads to the other, and then take the subway back to where you are staying.
Staten Island ferry is great, it is a good way to "breeze past" the Statue of Liberty and Ellis Island for free. Just take the subway to Bowling Green or South Ferry or Whitehall Street. You get on the boat, go to Staten Island, you'll be forced off the boat, then just hook around back thru the waiting area and back onto the boat to get back to the city.
Go to the top of a tall building. Two choices...the Empire State building (34th and 5th Ave.) or Rockefeller Center (6th and 50th Street). I think the Rockefeller Center is better, and the lines are shorter.
Take a cab ride over to Chelsea piers (23rd Street and West Side Highway)...and find a way to get into their gymnastics gym...it is absolutely amazing!
Go for a walk around any of the major parks. Union Square, Washington Square, Central Park, Tompkins Square Park, Battery Park...they are all fun. If you can afford it, take one of the horse and carriage rides through Central Park.
I love Little Italy...although Little Italy is getting littler every year. It is on Mulberry Street right above Chinatown. So many great restaurants and pastry shops and bakeries.
Take a walk along 5th Avenue, past all the stores decorated for the holidays. Take a walk through St. Patrick's Cathedral. And see the tree of course. If you don't mind waiting in line, go ice skating...its a lot of fun.
Go see a broadway show...there are so many to choose from. If your hotel can't get you tickets, go to the TKTS booth in times square. They sell same day discounted tickets. Get there early though!
Definitely get NY pizza. There is a pizza place on every corner practically. Its much better then domino's.
Go walk around the UN Headquarters. There is a reason why they call NYC the capital of the world...that is one of the reasons. You can walk from your hotel.
The financial district is kinda neat. There is a lot all in one area, the stock exchange, Trinity Church, Federal Hall, Ground Zero, and South Street Seaport.
Gosh there is so much to do....I can't even think of everything! If I think of more I'll post it!