I agree with everything sdsester said, just to have as a second opinion.Those boxes are to cover your remote valves. There are wires running from the controller (probably in the garage?) stations to these valves. The spickets were put in the boxes so you don't have to drag 100 foot garden hoses around.
The valves have rather standard components for closing off and adjusting water flow. You do this using the thumbscrew on the top of the valve to force it into "manual" model. The other component is a simple solenoid actuator that opens and closes the valve electrically from the controller box.
If the system has been sitting you should plan on rebuilding all the valves. Turn the water off and buy the kits for the models you have. The are just rubber gaskets, valve stem washers and so forth and do not cost much. they are easy to install.
With the valves rebuilt see if the controller turns them on. If not, you will have to replace the solenoid actuators (and attach wiring to them). The just thread on and off and are not particularly expensive either.
If rebuilding the valve seals does not keep that first box from filling with water? You may have a cracked pipe coming in or out of the valve.
Without seeing and knowing more of your system I cannot tell you for sure if you can bring it back to life. It sounds like it was originally installed properly so I suspect so. Plan on cleaning the whole system out including removing and reinserting the sprinkler heads and servicing the valves as mentioned.
If you are in a cold climate and the irrigation lines did not have water blown out at the end of each season you may have experienced some freezing damage that needs to be replaced.
Please send more details. Let's be optimistic and assume you can be back to sprinkling with neglected maintenance issues resolved.