A hydraulic actuator may have one thousand pounds of force applied to it. Opposing that force is a buildup of three thousand pounds of force. PSID is two thousand pounds of force on the high side, even though the system pressure is three thousand. It's the differential pressure.
From a practical application, where you will see it applied is in aircraft pressurization. To set cabin altitude, a certain amount of pressure inside the cabin must be applied. The higher you go, the lower the outside pressure, and the higher the differential pressure in the cabin.
At sea level, zero diferential pressure is required to hold a cabin altitude of sea level. At five thousand feet, however, 2.9 inches or so of differential pressure will hold sea level in the cabin. Or at ten thousand MSL, a 2.0 differential will hold five thousand feet, and so on. The numbers vary with the aircraft and the tightness of the fuselage...or the cabin porosity...how much air it leaks when pressurized outside of what the outflow valve can handle.
The exact numbers of PSID are relatively meaningless. For most aircraft, you merely set the values you want on the pressure controller, and the aircraft does the rest. If the aircraft doesn't do the rest, either abnormal or emergency proceedures apply...but the specific values are relative, and all you need to worry about is that they merely represent a value in the pressure difference between what's acting on the inside of the aircraft, and on the outside. A two PSID value means that two pounds per square inch pressure more are acting on the inside of the airplane pushing out, than outside pushing in.
Excessive PSID, or in other words, too much pressure inside the airplane, can damage it. I don't have it, but some sobering pictures are available of a KC-135 just a few years ago that got blown apart on the ground during maintenance, when the ground crew dummied the outflow valve and overpressured the pressure vessel. Too much pressure is just as critical as not enough.
Any time you see the "d" on PSI (eg, PSID), know that the author is attempting to communicate a difference in pressure between two sources or values, rather than merely an absolute pressure value. In this case, it isn't the pressure that's significance, but the difference between the two pressures that's doing the work, that has meaning, and which must be measured.