The concept couldn't be simpler, at least as far as mechanical systems go: use one type of kinetic energy to assist in moving a second type via fluid pressure transfer. The idea dates back further than anyone really knows, and the only thing that seems to have changed is the number of parts involved and the opportunities for failure that they bring. Still, it doesn't take a degree in fluid mechanics to diagnose the average power steering system. There are, after all, only so many things that can go wrong.
The majority of power steering pumps use a rotary-vane design, where small "walls," or vanes, connected to a rotor in the center push fluid through the system. The chamber in which the spinning rotor sits is oblong in shape; the vanes in the rotor actually slide into the rotor to go through the narrower portions and extend in the wider ones. While the rotary-vane design can adjust for a certain amount of wear, eventually the vanes will become so worn that they'll fail to extend fully in the wide parts of the chamber. The result is a loss of power steering pressure in the system and a subsequent reduction in power steering.
Power steering pumps are positive-displacement in nature, meaning that they'll always push a set amount of fluid as long as they're spinning. So, that may beg the question as to what happens when the system doesn't need fluid pressure to assist the rams in the rack. Excess fluid pressure in the system goes back into the power steering fluid reservoir via a spring-loaded relief or bypass valve. The bypass valve can malfunction in two ways. It can stick in the "closed" position and hydro-lock the system -- potentially stalling the engine -- or the spring can break or weaken prematurely. This will permit constant or premature fluid pressure bypass, which kills any chance the pressure has to help move your car's wheels.
The steering rack is a complicated mechanism that performs a simple task. When you turn the wheel, a rotary valve in the rack opens to route fluid from the pump to one side of the piston on the steering ram. When the cylinder in the ram pressurizes, it pushes the linkage and helps to turn the wheel. The rack can malfunction in a few ways, most commonly as internal leakage. If the pressure cylinder fails to seal, pressure will simply follow the path of least resistance back to the pump. Fluid valve failures are a bit less common, but do happen. Torque-sensing racks that use a torsion bar can experience malfunctions in the drag mechanism on the torsion bar. If it rotates too freely, the system will reduce steering assist to near zero under the assumption that you're not exerting much effort on the wheel.
Power steering fluid is an integral part of the system, just like blood is a crucial part of your body. It's so important that manufacturers all but engineer the system around the fluid and its chemical properties. A low level of power steering fluid is far and away the most common cause for loss of power steering, and always elicits that tell-tale shriek from the pump. Fun fact: That shrieking is the sound of tiny bubbles vaporizing and exploding in the pump, "cavitating" to produce high-pressure shockwaves that vibrate the pump and produce that ear-splitting whine.
Old, burned fluid will typically reduce steering assist and reduce component longevity, but won't usually result in a total loss of steering assist. Water in the system will vaporize quickly and act like trapped air. It would take quite a bit of water to kill the power steering completely, but even if you filled the system with nothing but water you'd probably still have some inkling of power assist. At least, until the pump explodes. Contaminants like rust, dust, rubber, nylon, dirt and squirrels in your fluid can easily clog any of the tiny passages in the system and cause a loss of power steering -- particularly if they happen to lodge in the bypass or steering rack valves.
