BobMcGee indicates that the thickness of the skin is determined by the handling or the variety. Although variety can determine the thickness, handling cannot make the skin thicker or thinner. Handling may cause the thin skin to come off, but it won't cause it to be thin in the first place.
The thickness of the skin within a variety of potato is determined by how long the potatoes were left in the ground after the plant dies. If the potatoes are harvested while the plant is still alive, they will have an exceptionally thin skin that will peel off easily. You can peel them by rubbing the surface with your fingers. This is almost always the case with "new" potatoes and is true across varieties from reds and pontiac to yellow yukon and russet.
Thicker skins result from leaving the potatoes in the ground after the plant dies (from a frost for example) for days to weeks. The potatoes stop getting nutrient from the plant and thicken their skins to insulate themselves from the elements. Producers allow the potatoes' skin to thicken because it improves their lifespan.
Thick or thin skins with a scaly texture have been infected with a fungus and should be peeled a little more deeply than usual. It is typically the result of the soil having too high a pH.