Furnas Probably Contactor
The Furnas box is probably a contactor and the black button is a reset device for the overload protectors.
Somewhere above the Furnas box there should be a circuit breaker or a pair of fuses or a disconnect switch.
Open the Furnas box (carefully) and check to see if there is Voltage at the inlet of that box. There should be about 230 Volts between a pair of wires, and 115 Volts between each wire and neutral (probably white or bare wire).
Note to help us avoid confusion - please use the term "current" only when talking about Amps.
If you don't have 230 Volts across the hot pair at the inlet to the Furnas, then your power is disconnected for some reason. If you have 115 Volts from one to neutral but nothing from the other to neutral then you have a power loss on one of the hot leads.
If you don't have the correct power, then find out why and restore power to the Furnas before going further.
If you have power at the incoming (usually top) of the Furnas, check for voltage at each end of the coil relative to neutral, and across the coil. The voltage on the coil, if there is any, is probably coming from the pressure switch.
Now the contactor coil may or may not be wired through the overload protectors in the Furnas. From an earlier post where you mentioned pushing the black button, I suspect that the coil is wired through the overloads.
Now we are going to try a little experiment. Using a screwdriver so your hands don't get too close to the hot wiring in the Furnas box, push and release the black button and see if the contactor pulls in. If it does it will probably startle you. See if the pump starts; have someone near where they can hear it if you can't hear it from where you are. If the pump starts when you pushed the overload button, then it probably means that the overload was tripped out from overcurrent in the motor, and you may have a pump failure. In that case it will probably trip out again pretty quickly.
Check it as far as this goes and get back to us.
Post an electrical schematic if you can, at least as far as the order of boxes in the system. We are just guessing on the alternatives if we don't know where all the controls and Pumptec and other things are connected in your electrical circuit.
If you are going to buy a meter, consider the extra investment to get a clamp-on ammeter which will also measure Volts and resistance (ohms). I have seen some reasonably priced one at Home Depot. I got a Sperry DSA-400 Digisnap there a few years ago which has served me well. It isn't practical to measure current in your type of system with the usual Volt-Ohm meter.