The only water fountain in the Mechanical Engineering Department at Rutgers is broken. It has been broken since the summer. This has resulted in the water fountains innards being exposed.
The water fountain seems to work exactly as air-conditioners work. The black container at the bottom is a compressor, which compresses a coolant gas. The high pressure gas is then run into the condenser which is the fan and radiator on the right side. The condenser cools the coolant gas that was heated by the compression and allows it to condense into a liquid. The liquid coolant is then pumped to a evaporator. I believe the evaporator is inside the cold water storage container, which is at the top of the box insulated by what appears to be Styrofoam. As the liquid coolant comes to the evaporator it is allowed to evaporate which results in a cooling process. In much the same way sweat evaporating cools you down. The gas is then pumped back into the compressor and the cycle repeats. The cool water then stays insulated and when you push the button is pumped right to you.
Here is a better image of the compressor and condenser. Additionally, there is a valve on the left side of the image, however I cannot recall if it is on the high or low pressure side of the cycle.