Arduino Nano For Home Automation 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.