The pinion (2) is attached to the steering shaft which is attached to the steering wheel. Thus, as the steering wheel is turned, the teeth in the pinion mesh with the rack (1) and slides the rack left and right. The rack is attached to the tie rods (3) which in turn attach to the knuckle (not shown) and tire. That's manual steering with a rack and pinion.
In the power assist system, fluid from the pump (high pressure side) enters the steering gear at (4) and returns (low pressure side) at (5). A valve system inside the gear box directs fluid out at (6) and (7) which re-enters the gearbox at (8) and (9). This pressurized fluid acts on the power piston (10) to assist steering. Thus, if the driver is turning right, additional fluid is directed to the right side of the piston, which helps move the rack and therefore steer the tires.
Fluid is directed to either the right or left sides by means of a rotary valve and torsion bar. Through this mechanism, the fluid pressure (and thus the amount of "assist") is proportional to the torque due to steering effort, and not the direction that the wheels are turned. In other words, it is possible for the system to help you turn left, even if the steering wheel is right of center. Here is a sketch of the rotary valve. Only one quarter is shown (it's symmetrical).