A central air-conditioner is an elaborate system that involves parts such as an evaporator, a compressor, a condenser, fans, and refrigerant to run through the unit. Like many household appliances, each of these components does their job both independently, but also in unison with the connecting parts and when one section of the system goes down, the entire performance falters.
While each component in a central air-conditioner is critical, quite simply the unit will not work if the fan is not running. Fans in the central A/C both pull air out of the room to be conditioned and have humidity removed and also push chilled air through the vents throughout the house. When the fan goes down there is no way to cycle the air and thus the system is essentially stalled. It’s important to get the fan up and running ASAP or else the home will get uncomfortable in a hurry and you could end up damaging the compressor.