These are what is called LV controlled lighting, I think GE was the main manufacture of them.
They are a very simple system each relay has two coils one for on and one for off and if you have the version to provide a light to let you know when light is on there would be a contact in the relay and a forth wire that would feed the pilot light in the switch otherwise you would only have three wires at the relay, the coils would flip flop the contacts on or off depending on which coil was energized, from the transformer one conductor would go to the common on the relay, the other would go to the common on the momentary push button switch's.
The low voltage wiring was usually 18/3 or 18/4 door bell wire, anything smaller would cause problems on long runs, if there is master switch's they would have runs from the relays/lights that they wanted the master switch to operate and it would have a rotary select switch above a set of off and on push buttons, many times master bedrooms and or family rooms would have a table top unit that connected to a wall jack through a milticonductor cable, that allowed the owner to control many lights without getting up, there is also motorized all on and all off units that would turn on/off every light in the house, or selected lights, also they had dimmers that were motorized with the actual dimmer in the main relay panel, hole the on button and the lights would get brighter, hold the off button and the lights would dim till they turned off.
I have installed many of them back in the W's when they were the thing, most had the relays located in a K/O on the box for the fixture, they snapped in from the inside of the box with the LEV connection on the outside of the box which being careful popping them out then slowly guide the LV wires into the box to change a bad relay.
Common problems in trouble shooting is bad relay (on coil or off coil), using a voltage meter at the switch from the common to each button would tell you if the on or off coil was bad assuming wiring was good, as you should see 24 volts or close to it across each push button which is measuring through each coil back to the transformer, if both coils were good then the switch is bad, also some times the mechanical part of the relay would go bad and both coils would show good, but having a person by the relay to listen for the relay to switch would check this problem, bad HV contacts were rare, but if you had a relay that you could hear the switching then check across the HV contacts to see if the contacts are closing, this was only if the light didn't come on.
I have worked on many different type of installs on these the hardest ones was where they installed the relay into one light that controlled a few others if the relay stopped working it was a guess at which one had the relay, can lights were a pain when they put the relay in the junction box of a can light with a few other cans on it also, as you had to hunt for which one had the relay, of course if you had a nice installer they would leave you a map to the relays in the main breaker panel, but don't bet on it.
The system you have with the relay panels is the easiest to deal with, have your helper or homeowner push the on or off button (depending upon which is working) and you should hear a buzz from the relay, fairly easy to identify which relay goes to which button.
Parts are still available, and most supply houses that deal in GE stuff can get the relays or buttons, or you can get them from several sites online, just search for
GE low voltage lighting control as in the link above which turned up many sites.
I have rebuilt many of these systems as it is in most cases cheaper to do then try to install wall switches to control the lights as in your case if fed all the way to the remote relay panel, my sisters house had the relays in the light box's but we still just replaced the bad relays as she like the convince of having the table top multi master controls in her bedroom and living room with the multi-master switch's at various locations through out the house such as the one in the kitchen that controlled the dinning room light, the hall, all the outside lights, which is one of the nice things, and the outside lights were controlled from many rooms, as well as from the garage, each bedroom and bathroom had two sets of switches one for the light in the room and one for the hall lights
