air conditioner compressor will not start or engage

True - the air conditioning compressor will not turn on when it gets no power - as with when a cycle switch has failed or a wiring connector at that switch, or perhaps a compressor coil is weak. However, it will also not engage when the system pressure is too low, as evidenced by low refrigerant level, which is caused by a leak. Have the refrigerant charge checked first - you will likely find it is low and start seeking out leaks.

If there is not enough freon in the AC system then the compressor will not engage.