From wikipedia, the purpose of water towers (in summary, since I doubt you will read the whole thing.)
-Water towers are able to supply water even during power outages, because they rely on pressure produced by elevation of water (due to gravity) to push the water pipes into domestic and industrial water distribution systems; however, they cannot supply water for a long time without electricity, because a pump is required to refill the tower.
-A further purpose is that a water tower serves as a reservoir to help with water needs during peak usage times. The water level in the tower typically falls during the peak usage hours of the day, and then a pump fills it back up during the night. This process also keeps the water from freezing in cold weather, since the tower is constantly being drained and refilled.
The second one more or less explains why there are so many in an urban and highly (highly) populated area like Manhattan.
EDIT: After scrolling down a little bit, I found this.
In the 1800s, New York City required that all buildings higher than six stories be equipped with a rooftop water tower. This was necessary to prevent the need for excessively high pressures at lower elevations, which could burst pipes. In modern times, the towers have become fashionable in some circles. As of 2006, the neighborhood of Tribeca requires water towers on all buildings, whether or not they are being used. Two companies in New York build water towers, both of which are family businesses in operation since the 1800s.
And there you have it.
