The analog input is really just "sensing" voltage. Very-very little current flows into the analog input. And of course, that voltage is there whether the Arduino (or a multimeter) is connected or not.
The potentiometer is a variable [u]voltage divider[/u].
For example, with the pot at mid-position you have two equal-value resistors and the voltage is divided in half. With a multimeter, you can measure the voltage across the top resistor (or the top half of the pot) or the bottom resistor. But, the Arduino's analog input is referenced to ground so you can only measure the voltage across the bottom resistor (or you could measure the 5V at the top, referenced to ground).
If you take an electronics class, the 1st thing you learn is [u]Ohm's Law[/u] which describes the relationship between voltage, resistance, and current. i.e. More voltage = more current. More resistance = less current. (Resistance is the resistance to current flow.)
At some point, you learn [u]Kirchhoff's Laws[/u] which describe how voltages divide in series circuits and how currents sum in parallel circuits.
As long as you realize that resistances add in series and that the same current flows through both series resistors, you can simply apply Ohm's Law to calculate the current, and use Ohm's Law again to calculate the voltage drop across each resistor in a voltage divider.
It's important that very little current flows into the Arduino (or whatever else is connected), otherwise the current won't be the same through both resistors (or both halves of the pot) and the calculations are not quite as simple.