PWM: Pulse Width Modulator, a DC motor speed controller that maintains torque, like when you run a cordless drill at low speed you can hear a high pitched whine from the PWM.
Gain:The fan speed for a particular temperature, ie at 60C fan runs at 20%, at 70C fan runs at 60%
Derivative: (Rate) Looks at the rate of change in temperature, so if temperature is rising fan speed increases above what the gain would have set and slows the fan as the temperature.
Integral: (Reset) Increases the fan speed when a steady state higher temperature is detected, so if temperature was 70C the fan would slowly ramp up to 100%(until the temperature starts to drop then the gain & derivative kick in and slow the fan).
Similar to driving a car, we push the accelerator to accelerate then start to lift as we approach the speed we want, then if the car slows we push the accelerator more and more to maintain speed, then lift off as the car exceeds the speed we want.
Graham