Like on the show how the day before they actually draw the tattoo, they will draw the design on a piece of paper, wet the paper, then stick the paper on the person and see if the person likes how it looks. Then the next day they do the real tattoo. My question is how do you make the thing on the piece of paper.

It’s a special paper called Thermal Paper. Any tattoo supply company will stock it and you can find it on ebay if you search for thermal tattoo.
http://www.pricksofthetrade.com
Or just type in: Tattoo Supplies