user8046: 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.

user3671: 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. 

8046: <p>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.</p>
 3671: <p>A relay is a common way to allow a smaller power source to safely control a larger one. A relay is like a switch.</p>

<p>Here's some <a href="https://learn.adafruit.com/adafruits-raspberry-pi-lesson-13-power-control/hardware" rel="nofollow">reference material</a> 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. </p>
