Most residential power in the US is provided as single phase 220V with a center tapped neutral. This means that you can draw 220V between the two hot legs, or 110V from one leg to neutral. So long as the neutral is properly connected, it will keep the voltage on each leg approximately correct. (bextech, the neutral carries the *difference* in current between the two hot legs. If the two legs are each drawing the same amount of current, there is no current carried by the neutral. As soon as they fall out of balance, the neutral carries as much current as is needed to keep the two legs in balance. I can dig out the mathematical explanation if you'd like.)
Take away the neutral's connection to the power company's transformer, and you don't have anything holding each leg to 120V. The one that needs more power will actually get LESS, and the one that needs less power will actually get MORE. This is a Very Bad Thing. If this happens, 220V appliances will continue to work fine, because they are drawing hot to hot and not using the neutral. (This is what yuandrew described exactly.)
So why don't we use hot/hot all the time? Because it's more expensive. Center-tapped single phase is a very efficient way to provide 110V service while still having 220V available for heavy loads where the wire size required at 110V would be even more expensive. On ships, for instance, where there is no ground reference to tie neutral to, delta power (hot/hot) is used. That would be two 60V hots together forming a 120V supply, just like the two 110V hots in residential service together form a 220V supply.
One more complication - you're in NYC. Many older apartment buildings actually use commercial three-phase power. With that setup, you have a neutral and THREE hots that are out of phase with each other. You can develop 110V from any one hot to neutral, or 208V between any two hots. The advantage of this system is that it's much more efficient when all three hots are attached to a large motor. For instance, elevators work much better on three-phase power.
Note that I am talking about US/Canada power here. It's done differently other places; the UK, for instance, has a particularly unusual approach.