Having lived in New York City for about 3 years now (written July 2005), I've found some things that are unique to New York, which don't get mentioned often. I should preface this by saying that I actually live in Manhattan. I can't vouch for these things in other buroughs.
Subway and Bus Maps - the subway and bus maps that you can get for free at any subway booth are printed over an actual map of the city. This means that they can double as maps for the city streets, and also let you know exactly where a stop is in relation to where you're trying to go. As an added bonus, the bus maps (but not the subway) are to scale, so you can accuratly use them to guage walking distances.
Dipping your credit card / ATM - Machines that take credit cards and ATMs (even at the bank branch) do not actually take the card away from you. I always have a slight anxiety that the ATM machine will jam when trying to return my card. This is solved in New York because you just swipe the card, and put it back in you wallet. This way, you also have no chance of taking the cash and leaving without your card.
Free Furniture - As strange as it sounds, you can get high quality furniture for free on the streets of New York. Because of the difficulties of transporting funiture to a friend, or second-hand store, people will often leave their old furniture right on the sidewalk for the garbage truck to pick up. This is sometimes in good condition, and has simply been replaced by something even better. If you're patient, and keep and eye out, you're likely to run into what you're looking for. I've found chairs, shelves, and drawers that were being thrown out. You just have to get to it before the rain, and find an item that is still clean because it's just been put out.
Fireflies - Fireflies can be seen in park areas throughout the city during dusk in the summer.
Flowers from the corner store - In New York, the corner stores often have flower shops set up outside. This is very convinient, and the flowers are cheap compared to traditional flower shops. You don't have to think twice about bringing home a bouquet for $5 on the weekend.
Stroller Boogie Boards - Strollers have a little board on two wheels, attached at the back. This is so that the 3-5 year-old can stand and go for a ride while the baby is in the stroller, and the mother is pushing them both.
Delivery tupperware - Many restaurants (most noticably Chinese) deliver food in reusable plastic containers. These are quite useful for taking to work or cleaning and reusing.