I am sure that this are very basic problems, so I am also happy if you have something basic to read for me. I wondered the following: I have a 12 V supply and my Arduino. Now I thought when I have direct current, I have Gnd and + poles. So Gnd on Arduino and on supply should be the same. But when I use a Volt Meter and measure between + of 12 V supply and Gnd of 12 V supply, I get 12 V. But when I use Gnd of Arduino I get something in the mV range.
Can you explain me why this happens and Gnd != Gnd?
A piece of wire has finite though low resistance. The low current flowing in the wire between the two grounds by ohms law wil give a small voltage. A few mV is of no consequence in most cases.
If it worries you, just get thicker (lower R) wire.
You do have the two GNDs connected, right?
And I assume you mean that "when I use the ground of Arduino" you are measuring to the +12V. Right? Otherwise, related grounds SHOULD measure in the low, low, low mV.
GND on two different devices are not necessarily at the same potential. Just because there is a "common" reference on two separate circuits in no way means they are electrically related.
You need to know:
Is the 12V supply GND referenced (connected) to earth ground?
Is the Arduino GND referenced (connected) to earth ground?
If the answer to both of these questions is "Yes", then your ground points are referenced to each other. If you can't measure from Arduino GND to the +12V and get +12V then there is another problem.
If one or both answers is "No", then your GNDs are not the same. In this case, one or both of your GNDs are "floating". You can fix this by connecting the two GNDs together with a wire. Now you should be able to measure +12V from any GND to the +12V terminal.
If both GNDs are floating and you measure from Arduino GND to the supply +12V, you are making the only connection between the two with your meter. The meter will likely "pull" the +12V close to its GND (or visa versa) point because both circuits are floating with respect to each other. That's why you read mVs.
If the 12V supply or the Arduino alone was referenced to earth ground and you have connected the two GNDs together with a wire, be aware that both GNDs are now referenced to earth ground. If you have been probing around with a scope, and putting the scope GND at random places in the circuit, YOU CAN'T DO THAT ANY MORE. The scope GND is earth ground. It is dangerous to have two separate earth grounds in a circuit. (Mostly just dangerous to your circuit.) If both GNDs are floating, and you connect them with a wire, that practice MAY be OK. You have to know what you are doing if you want to keep having a scope. BUT DO NOT GO NEAR THE LINE SIDE OF THE POWER SUPPLY WITH THE SCOPE GND.
Perhaps you can explain one more question to me: I would have expected that one of the two supplies (Arduino 5V, Supply 12V) would deliver a different voltage if I edit GND. You understand my problem? Because Voltage was not 5V when I measured from Arduino + to Supply Gnd, Gnd is not the same. When I connect them, Gnd is changed - at least for one of the two supplies.
My explanation would be: Voltage is always generated relatively to the current Gnd in a supply.
I'm sorry, but you description is not clear to me. I don't understand what "edit GND" means. Are you using a simulation tool or physical components?
If your GND from Arduino and the GND from the 12V supply is connected, you now have one, single, common GND.
All of your measurements should be referenced from that one GND. There are multiple physical places, some on the supply, some on the Arduino, you can access that GND. So, yes, you are correct, all voltages must have a common reference and all currents return to their respective sources through the GND.
A similar situation you may have seen is a lineman that flies in a helicopter up to a high voltage transmission line. He holds out a wand and slowly approaches the line. An arc forms and charge is exchanged until the man is the same voltage as the line. He can now safely work because he is "referenced" to the line. As far as he is concerned, he and the line are electrically "on the same page". While this is not exactly the same as connecting your circuit grounds, I pick this example for the dramatic effect. Connecting your circuit grounds makes both circuits electrically "on the same page".