Helmets are always advisable!
Because many people ask this all the time, here is the official word from the NYC DOT web site:
In New York State, all bicyclists under the age of 14 years old are required to wear safety certified bicycle helmets when they are operators or passengers on bicycles (Sec. 1238(5)). Children aged 1 to 4 must wear certified bicycle helmet and ride in specially designed child safety seats.
In other words, cyclists or passengers on a bike who are 14 and older are not required by law to wear a helmet. Under 14, you must.
Some other helmet facts:
* Helmets (regardless of price) should have a sticker inside stating they have passed national safety regulations. Expensive helmets usually cost more because they are lighter (less fatiguing on long rides, easier to look around at traffic, etc.), have better ventilation (cooler in hot weather), are more comfortable (better cushion pads and strap systems) or might be considered more attractively designed.
* Helmets have a shelf life. The date of manufacture is usually stamped inside. If your helmet is more than 2-3 years old, you should replace it. If your helmet suffers a considerable knock (this includes dropping it really hard on the floor from a balcony, down a flight of stairs, etc.) you should replace it, regardless of its age.
* Helmets are useless if they do not fit properly, are not worn properly (people seem to wear them backwards sometimes) or adjusted incorrectly (again, better helmets usually have better strap adjustments).
* The DOT gives away helmets free to the public several times a year. They offer children's and adult sizes. Check their website for dates in your area.