Noise by noise
Squealing or screeching
Typically a worn bearing in the blower motor, or on older equipment a slipping belt. Not an emergency, but it is the sound of a part wearing out. Getting it handled before it seizes is cheaper than after.
Grinding
Metal on metal. A motor bearing has failed and the motor is destroying itself. Shut the system off. Continuing to run it usually turns a motor replacement into a motor plus whatever else it damages.
Banging or clanking
Something is loose, broken, or making contact with something it should not, often a fan blade or a component inside the compressor. Shut it off and call. This is the noise most likely to end in a destroyed compressor.
Rapid clicking at startup
The system is trying to start and failing, usually a relay or a failing capacitor. Capacitors are one of the more common and more affordable failures we replace. Left alone, the repeated start attempts stress the compressor.
Buzzing or humming
Generally electrical: a contactor, a loose connection, or a motor drawing current without turning. Sometimes it is just loose panel screws vibrating, which is harmless. Because the causes range from trivial to a genuine fire risk, it is worth having looked at.
Hissing or bubbling
Hissing often means refrigerant escaping. Bubbling or gurgling can mean refrigerant with air in the line. Either way you have a leak, and running low on refrigerant risks the compressor.
The short version: squealing, clicking and buzzing mean book a visit soon. Grinding, banging and hissing mean turn it off now. The cost difference between calling at the first noise and calling after the compressor fails is usually thousands.
Noises that are perfectly normal
A click when the system starts and stops, a low steady whoosh of air, and a brief whir as the outdoor fan spins up are all fine. So is occasional water dripping or trickling at the indoor unit, which is just condensate going where it should. What matters is a new noise, or one getting louder.