The electrical items in the car - ignition, starter, Iights, wipers and, of course, the engine management computer, all need electrical power to work. Without the alternator belt, you are relying entirely on the battery, which will keep the systems powered for a while. How long? Depends on how much stuff is switched on and how much charge is left in the battery. I'd replace that alternator belt ASAP.
If that belt also drives the water pump, you might only have a mile or two before it's game over.
A stitch in time saves nine.