There is an urban myth or perhaps wives tale that potato peels or "skins" are the most nutritious part of the potato. I decided to look into this a bit.
First of all, the starchy center of potato is not very nutritious from a vitamin and mineral point of view -- although it has good energy value -- but most people don't see that as advantage. The peel and the cell layer just under the peel are more differentiated, and has a wider variety of compounds. Some healthy and some less healthy.
The main effect of cooking a potato in the skin is that it prevents vitamins and minerals from being leaching out. 22% of vitamin C is leached away when boiled in the skin as compared to 42% loss when boiled without the skin. One can always not eat the skin when it gets to your plate, to avoid the less healthy compounds, and the starchy flesh is more nutritious.
The skin contains some minerals like potassium and calcium at higher levels than the flesh, but it has anti-nutritional ingredients like phenolics, glycoalkaloids, and nitrosamines. By anti-nutritional I mean chemicals that can have upsetting or even toxic effects.
The greenish skin of potatoes is unhealthy, and many recommend it be cut away and discarded. It contains solanine, which is a toxic glycoalkaloid. The structure is at right. You can see like the name says, sugar residues and a big alkaloid. Typically a potato has 100 mg/kg of solanine, but green potatoes have 1000 mg/kg, and the potato peel itself is 100 times more concentrated than the potato/peel mixture.
On the other hand solanine is not the worst toxin. It causes diarrhea, fever, and cramps, but can cause coma too. It affects calcium and potassium transport in cell membranes. One population study showed a correlation with the birth defect spina bifida, but this has not been confirmed. Green tomatoes are a much more serious source, and green tomatoes should never be eaten raw. Solanine has a bad flavor, and people avoid foods high in solanine based on taste alone, unless they are starving.
A 25 mg dose has been taken as a practical threshold for nausea. This would be about 25 g, or perhaps a large tablespoon of green potato. Strangely this is only 250 g of standard potato, so this must happen fairly routinely, and or the 25 mg dose number is not that meaningful or valid.
The potato peel does not take up much of the mass of the potato, so its nutritional value is almost undetectable. This is from a New Zealand potato board study. Based on this, there is little difference if the potato is peeled or not. For example there is a two calorie difference -- more for the unpeeled.
In conclusion, people should not eat the skin. Very little advantage and why eat these potentially harmful substances?