Darlington

A small signal transistor can be activated by a small current flowing through from its base connection to the emitter pin.

The voltage difference between the base and emitter needs to rise to about 0.6 Volts for the transistor to be fully activated. If the power of the input signal is weak, (power = Volts * Amps) and it is needed to control a heavy load current (for a DC motor) the small signal transistor, by itself may get to hot or burnout when the current flowing through the collector and the motor is running.

To up-size the power of the motor which will need more current, we need to switch it with a device that can handle a greater current flow. A typical device would be a "power transistor". In this diagram a TIP31C transistor is used. 

The power transistor in the above diagram provides a current shunt for the speed control Potentiometer. The specifications say that the power for the pot' is 200mW.  that would burnout if it tried to regulate all of the current that the motor required.

The Darlington pair transistors are working with a large capacitor.

Need to add a zener diode to stop the capacitor from charging over rated voltage.