The led behaves like a diode with a forward voltage of around 1.7v...1.9v (for a red led like in the picture) but this diode only conducts once there's some amount of current flowing through it.
There is also a very narrow region in which the led behaves like a resistor, it's not fully closed and it's not fully on.
So when you have the potentiometer in the circuit, you practically have a resistor in the circuit which limits the current flowing through the led. If the resistance is small enough, enough current flow that the LED is turned on fully and there's a voltage drop across the led.
You are measuring the voltage drop between the led terminals with your multimeter.
If you adjust the potentiometer, you're either reducing the current flowing through circuit too much, so the led starts to turn off, or you're letting more current flow and you could either damage the led or the batteries could suffer and lower their output voltage, if they can't supply the demanded current.
if you use a 0 ohm resistor, you're measuring the voltage drop between the multimeter probes. Since the resistor is 0 ohm, the resistance of the wires is very small and the current is very small (because it's limited by the potentiometer) then the voltage drop is practically 0.
Remember Ohm's law ... Voltage equals current x resistance .. you can use it pretty much everywhere.