I would like to use my arduino pins to control a 12 V light that has its own power supply. So what I need is something that can be used as a switch (here is in pink color) but I don't know much about electronics so I don't know what part to put there. I don't want to use transistors as they require to use the same ground potential.
For simplicity and robustness, use a relay. That's basically a electrically-controlled mechanical switch.
You can get modules that contain relays and that are controlled directly from digital outputs. You can also make your own circuit to do that. Driving a relay from a digital signal has been discussed many times here before. Look around.