The answer to 'why is my air conditioner not turning on' is that there could be several common causes, including a tripped circuit breaker, a thermostat issue, a problem with the compressor or electrical components, or low refrigerant levels. The symptoms you may notice include the fan not running, the compressor not kicking on, or the unit not blowing cold air.
If the air conditioner is not turning on at all, the first thing to check is the circuit breaker. Make sure it has not been tripped and reset it if necessary. You should also check the thermostat to ensure it is set to 'cool' and the temperature is set lower than the current room temperature. If those basic checks don't resolve the issue, it's likely there is a more complex problem that will require the expertise of a professional HVAC technician.