My question is about the arduino Nano or Micro and the smallest and cheapest way to control a bigger power source. I would wire it with a socket so that I could send a command from the board and a light would turn on for example.
A relay is a common way to allow a smaller power source to safely control a larger one. A relay is like a switch.
Here's some reference material about using a relay with a microcontroller. You shouldn't have much trouble finding more material now that you know what you're looking for.
So your question is: "How to control a bigger power source using the Arduino Nano or Micro".
This tutorial on youtube gives an explanation: https://www.youtube.com/watch?v=sOz41WQF7wE
It's about a motor, but the answer is the same. You should use a relais/mosfet
This one is actually pretty great also: https://www.youtube.com/watch?v=Te5YYVZiOKs