Here, we explain the general concept of using an operational amplifier for voltage regulation. By utilizing an op-amp and few other external components, we can easily build a linear voltage regulator. Apart for being a regulator, the same circuit is also a voltage stabilizer, able to stabilize voltage at a grade better than 0.01%. The circuit is powered from a non-stabilized DC-power source, and uses a transistor (T1) inside a feedback loop. The transistor is used to supply the load with much more current than the op-amp itself could possibly supply. The D1 diode is a Zener - type  diode and it is used for voltage reference.

Basic op-amp voltage regulator
Basic op-amp voltage regulator

D1 is biased through Rz. When correctly reserve biased, the Zener diode keeps the voltage across its leads close to the Zener breakdown voltage. The op-amp is used as a linear voltage amplifier. Due to the high open loop voltage gain of the op-amp, and as far as the op-amp remains in its linear region, the voltage difference between its inverting (V-) and non-inverting input (V+) is almost equal to zero. In other words, the voltage at its non-inverting input, in respect to the ground, equals the voltage at its inverting input: 

V-=V+    (1)

Equation (1) holds true for any op-amp working at its linear region (as an amplifier).  

R1 and R2 form a voltage divider, and the voltage (V-) at their connection point is also given by the well known voltage-divider formula: 

V-=VL·R1/(R1+R2)     (2)

However, V+ is also equal to the Zener breakdown voltage (Vz), because the non-inverting input of the op-amp is directly connected to the cathode of the Zener diode.

V+=VZ (3)

After solving (1),(2) and (3), we get:

VL=VZ·(1+R2/R1)    (4)

From equation (4), we conclude that VL voltage (which is the voltage applied to the load) is directly proportional to the Zener voltage. As far as the Zener voltage remains stable, VL also remains stable. Additionally, the voltage applied to the load, can be easily adjusted by adjusting R1, R2 or both of them. For continues voltage adjustment, R1 and R2 should replaced by a potentiometer, having its wiper at the non-inverting input of the op-amp, and its other leads at the ground and the VL line, respectively. 

VL, is not possible to exceed VDC. It can be almost as much high as VDC when T1 saturates, but no more than this. VL (the voltage at the load) could not also be lower than Vz. That's why VZ<VL<VDC.

As in any linear regulator, heat losses on T1 increase when the output voltage decreases. In fact, the power loss due to heating is the current times the voltage dropped across  T1. Besides heating losses, a linear regulator is often preferred over a switching one because it does not require any inductors which can be relatively expensive or bulky.