In your house the neutral should be connected to earth ground at the breaker box. With no current flowing there is no voltage on the neutral (relative to ground). When current is flowing the slight resistance in the wire can create a slight voltage drop across the wires, and the neutral may not measure exactly "zero".
The current flows through a "complete circuit" through the hot & neutral wires and through the load. Under normal conditions, no current flows through the ground wire. If you "break the circuit" but turning off a switch, unplugging an appliance, or if you cut/disconnect either the hot or neutral, no current flows.
Voltage is always present at the power outlet (assume no associates wall-switch) but current only flows when something is connected when something is powered-on. Power (Watts) is the product of voltage & current (Volts x Amps).
The AC (alternating current) follows a sine wave from zero up to the positive peak, back through zero to a negative peak, and then back to zero again. Assuming you have 50Hz were you live, that cycle repeats 50 times per second. ...The AC voltage is positive half the time and negative half the time. And of course, the direction of current flow also alternates at 50Hz (or 60 Hz here in North America).
The Arduino ground is usually just a reference and "ground" is a "concept". For example, if your Arduino is battery powered the battery's negative terminal would be connected to the Arduino's ground, but there may be no connection to earth ground. In this case, current does flow through the ground wire.
If your Arduino is connected to USB and you have a desktop/tower computer, the Arduino's ground will be connected through USB, through the computer, and finally to the wall socket's earth ground.
The Arduino can be damaged by negative voltages (relative to it's ground) but sometimes (often with op-amps) you need a dual power supply (positive and negative voltages). If you connect a 2nd battery with it's positive terminal to ground, the negative terminal of the battery becomes your negative power supply.