All the electric gauges use a float of some sort which operates a variable resistance, or rheostat.
One of the problems in measuring fuel level is the fact that fuel sloshing about will bob the float up and down, giving erratic readings. Two schemes are used in VW gauges to overcome this.
In one, the float is encased in a vertical cylinder with only tiny holes allowing fuel to pass into the cylinder. Thus, even if the fuel level outside the cylinder varies rapidly, the level inside remains relatively static.
The gauge is a simple voltmeter style and responds rapidly to changes in float resistance.

In the second scheme, the float is allowed to bob up and down at will but the signal is damped by the gauge mechanism. In this case, the float resistance is used to heat up a bi-metal arm inside the gauge. As the float moves toward Full, more current flows and the bi-metal gets hotter, thus raising the needle's position.

To improve the accuracy of the bi-metal type gauge, a form of voltage regulator is used to supply current to the gauge. The most common type used was a mechanical vibrating contact which alternately turned current on and off going to the gauge. A bi-metal inside the regulator made it responsive to the incoming voltage. This regulator (or stabilizer, or simply vibrator) was mounted on the back of the gauge.

How To Test: In case of mal-function it's important to be able to tell ff the problem lies in the tank sending unit or in the gauge.
1) Disconnect the sending unit wire. The gauge should move to Empty.
2) Short the sending unit wire to ground. The gauge should move to Full.

If it passes both tests, the gauge itself is working correctly.

If it does not pass, the sending unit may be defective.
Other possibilities: If the gauge does not move at all, the fuse supplying the circuit may be blown, or the vibrator may be defective.
If calibration is off, the vibrator could be defective or the sending unit is sticking or defective.

Erratic gauge readings can be caused by a worn out sending unit or, possibly, by loose terminals on the gauge. Loose terminals were quite common on some model years.

If you have to remove a sending unit, use extra caution. Some units twist into the tank with a bayonet-style lock. Use tools which won't create sparks when removing. In all cases, keep any flame away and don't use a "trouble light" which could ignite fuel. Disconnect the battery before doing any work on the fuel tank.