Does anyone know the name of that special pen that あなた can use to draw on pictures on a computer?

Since I have one I can answer this.
A stylus pen, they come with tablets.
I got both of mine from a friend, their old and beat up. I need a new one but I don't have the money yet. XD

A Sharpie marker will work if あなた want your drawing to be a permanent part of your computer monitor, even when あなた turn it off. But for non-permanent doodling, use a stylus on a graphics tablet.

it's stylos, I guess.

Bamboo tablet または something like that.