e.g. the plant is a DC PM motor preceded by a power amplifer. Motor angular shaft position is measured by a potentiometer and fed back to the controller where it is compared with the required angular position, r. The PI controller provides the appropriate control law to meet dynamic performance specification. But note that a position control loop has an inherent integration, therefore doesn't really need the I in PI.
As above, but replace the potentiometer with a tachometer that measures motor shaft angular velocity. This produces a velocity control system where the required velocity is the input, r. This time there is no inherent integration, so if the controller does not have an I term there will be a necessary error signal to drive the motor at a constant speed (otherwise friction torque losses will slow it down). The integrator in the PI controller means that the motor will run at a constant speed with a zero error signal (as long as there are no transient external disturbances).
The plant is a process that must be maintained at a constant temperature. Plant output is actual temerature that is measured by a sensor and compared with the required temperature, r. A temperature control loop normally requires the controller to have an integral term in order to reduce the error to zero, otherwise the system will operate with a constant error signal to offset temperature loss due to normal convection/conduction heat losses. The P and I gains are adjusted to obtain the required performance.
In crystal-pulling by the Czochralski method (e.g. to grow cylindrical silicon semiconductor crystals) the rate of change of crystal weight is controlled, as it is pulled from the melt, so that growth is optimised for quality of the crystal lattice. In this case the plant is the crucible/melt/crystal. Measurement of output is the crystal weight which is differentiated and compared to the required rate of change of weight, r. The controller is tuned to optimise growth conditions (although the control law for this example is much more demanding than a simple PI or PID can handle without an adaptive controller that overseas the entire process, which is inherently unstable)