The Proportional Gain (Kp) works in the present. It acts on the error right at this particular moment. The Integral Gain (Ki or Ti) works on the past history of the error. The integral portion of the PID regulator contains the history of torque required to get the speed right.
If required, the Derivative Gain (Kd) monitors the rate of change and anticipates and corrects for the future.
Speed and or tension regulators are tuned in the order listed above. The proportional gain is set first taking care of the immediate large error in speed or tension. Then the integral gain is set. This takes care of the steady state error which requires knowing the control action required in the past to keep the error near zero. If a large overshoot is a problem, the derivative gain is set. This anticipates that an overshoot is about to happen and compensates for it.