Significance
Circuit breakers are designed to protect wiring, equipment and you from potentially dangerous shorts and overloads. If something goes wrong in the circuit, the breaker trips, alerting you to the problem. A breaker that's emitting a continuous buzz is on the verge of tripping. That means there's a problem of some sort, it just isn't bad enough to trip the breaker yet.
The Circuit Breaker
The first thing to consider is that the breaker itself may be weak or faulty. While that may be a relief if you were fearing the worst about your air conditioner, don't shrug it off. A malfunctioning breaker can put your expensive equipment at risk or cause a fire.
The Wiring
If any of the wiring in the circuit feels hot to the touch, trip the breaker yourself and call an electrician or A/C repairman. Otherwise, look for loose connections, discolored insulation on the wires, pest damage or any other signs the wiring has issues.
The Air Conditioner
If the air conditioner isn't functioning efficiently, it draws more power. Check the inside portion of the unit for a dirty filter, dirty or clogged evaporator coils or blocked intake ducts. Look at the outside section for blockages of the cooling fan or condenser coils. Listen to the unit as it starts. Some compressor motors are noisier than others, especially older models. But it's usually obvious when a motor is having trouble starting or running. Installing a hard start capacitor kit might help the air conditioner and solve your dimming lights or buzzing breaker.