Potato "eyes" are the sprouts that form on the surface of the potato skin that eventually develop into the above ground stem or the stem of the tubers. If you're trying to grow potatoes, then eyes are a good thing, since they'll sprout leaves and form part of a "slip" - a rooted cutting off a potato. If you aren't trying to grow potatoes, why not? Everybody should be growing potatoes.
But green potatoes, or potatoes whose eyes have sprouted may have alkali concentrations that make them taste bad or even mildly toxic. I don't peel potatoes (not even in my pocket) but I do peel or cut out the eyes.