I posted this in another thread but will repost here just for continuity.
Technically speaking, power beyond is a feature on a directional control valve that allows the pressure gallery to be isolated from the tank gallery and be carried over to an additional valve - usually another directional control valve.
The implementation of the power beyond feature varies with valve type and manufacturer. However the most common arrangement is to install a threaded plug or sleeve that blocks the internal connection between the pressure and tank galleries inside the valve. This creates two outlet ports, the power beyond and the return to tank. The power beyond port is then used to supply high pressure pump flow to the additional directional control valve. The return to tank port returns neutral pressure flow from the work ports (from the cylinders) back to the tank reservoir.
That's the technical description and is an edited version of what is available from "The Insider Secrets to Hydraulics".
In simple laymans terms, if you have more than one valve in a series hydraulic circuit, all valves except for the last must have power beyond capability since neutral pressure exhaust fluid from the valve work ports cannot drive the next valve in the circuit.
When a spool is activated flow is redirected to the cylinder via one of the work port connections. Exhaust from the cylinder is then emptied into the tank gallery thru the other work port connection.