This is my first post so excuse my explanation if its a bit too sloppy.
The answer to you question is "both depending on how the component is wired".
If you wish to understand how a potentiomenter also called a pot works try taking a breadboard and wiring several resistors 1Kohm in series. Like a voltage divider but with several resistors instead of two. Apply 5V across this series combination and then measure the voltages between, the negative of the power source and the points between resistors. You should see how the voltage changes. The voltage divides proportionally across all the resistors. Why is this? Because of ohms law that is. When you apply the 5 volts, current flows through all of the resistors in series and a volt drop occurs across each one. What you are measuring is the sum of the volt drops at different points in this chain of resistors.
Now remember any resistor can be modelled as a sum of resistances in series. For example a 10K resistor is technically the same as 10 1K resistors in series. Now imagine you took a piece of carbon or graphite like a pencil and took the graphite rod out. That length graphite rod is a resistor. If you take an ohm meter and measure you will see the resistance halfway down the length is approximately half the total resistance of the full length. If you measure a quarter of the piece of graphite you fill find the resistance a quareter of the full length. You could keep measuring like this ad infinitum and you will eventually realise that you can think of this graphite rod as being made up of millions of tiny resistances in series.
If you apply a voltage a current flows causing volt drops across all these resistors. If you then take a voltmeter and measure between one end and the mid point you will get half the supply voltage. If you measure between one end and a quarter of the way up from that end you get a quarter of the supply voltage. If you slide the one voltmeter probe up and down the graphite rod, while keeping the other probe on one end you will see the voltage varies and gets bigger as you move the probe further from the end where the other probe is. You now a crude potentiometer. The wiper slides up and down the graphite rod which is connected to the power supply.
If you can get those non multiturn trim pots I suggest you try and break one open and you will see a graphite track with the wiper moving across the track.
So you can think of a pot as a variable voltage divider of sorts.
If you want to control current you need a variable resistor or rheostat, a pot can also be used as a rheostat by only using two of the terminals. You use the wiper and one of the other terminals. I leave this for you to think about.
Pots are often used in microcontroller circuits as a simple way of obtaining an analog voltage. If in other circuits pots are actually used for this purpose. There are limitations such as being careful not to load them too much, but nothing you have to worry about when using Arduino 