What activates the contactor relay on a central air unit.

In most cases there is a 24 volt transformer on the circuit board that supplies the contactor - the thermostat actually activates the contactor through the 24 volt circuit. Your problems could possibly be due to a faulty thermostat, circuit board, contactor or any of the low voltage wiring that connects them all up.

My contactor on a Heil unit was a honeywell 24v was calling and power was at the contactor. The coil was hot but no noise. I removed one of the 24v leads and the contactor started working. Time for replacement.