My mother's home has these metal panels in the closet on the second floor, directly under them on the 1st floor is the breaker box. The house has push buttons for the lights instead of switches. There is no breakers under these panels and everytime a light is turned on or off there is a click. I am writing a paper and I need to know what these boxes are called.
Most likely the boxes contain relays that are controlled by low-voltage circuits from the push buttons. A relay is an electrical switch that is controlled by an electrical circuit. In this case, the relay allows low-voltage wiring and push buttons to control the light circuit.
This setup was sometimes used before modern home automation systems existed to allow control of a circuit from multiple locations or with devices that were not rated for normal line voltages, and a variation of it is sometimes used today where the switches are located near the circuit panel and remote switches are located in convenient locations.