Turn on/off laptop screen

I'm having trouble finding any examples of how to control a laptop via arduino. My goal is to be able to turn on or off the laptop's screen, to start with. I imagine it will have to be done through serial commands.
I apologize for being vague, my unfamiliarity with laptop control (and inability to find similar programs to reference) makes it hard to be specific. I appreciate any advice, thanks!!

You will be better served with an Arduino Leonardo. It can emulate a USB keyboard connected to your laptop and can therefore do anything you can do from the keyboard. It can also emulate a USB mouse and therefore do anything you can do with a mouse.

If you want to use a serial interface you will need to write the program on the laptop that accepts the serial communication and does the desired actions. You will have to design a protocol for the actions you want to implement.

I was hoping there'd be a universal way to tell the computer to turn off the screen (so that the same commands would turn off varying types of laptops, at least on the same OS platform). I'm not familiar with programming outside of arduino though, so if you don't think there's an alternative then I'll see what i can do to make your suggestion work best. Thanks again, i have a start point now!

at least on the same OS platform

If this is an Apple then you can use Apple script to receive serial from the Arduino and turn it into OS actions.