Whenever peeling a fruit or vegetable, someone will usually comment (sarcastically), "that's where all the good stuff is", meaning vitamins, minerals, or whatever. Do the skins or peels of common plants in Western foods (apples, potatoes, carrots, pears) really contain the majority of the nutrients? They would seem to contain a bit of fiber, but what about lighter compounds?
Let's first define "nutrients". There are macronutrients and micronutrients.
Macronutrients are the familiar protein (amino acids), carbohydrates (sugars and starches including fiber), fats (saturated/unsaturated/polyunsatured and fatty acids). These are the substances that provide energy and building materials. An alternative definition of macronutrient is the set of chemical elements we need the most of in large amounts: CHNOPS (carbon, hydrogen, nitrogen, oxygen, phosphorus, sulfur). Additionally water is an important bulk nutrient.
You can't survive without macronutrients, and they are undoubtedly "good stuff". So in an apple, for example, by far the bulk of the macronutrients are in the flesh and not just the peel: water, sugar, fiber.
Micronutrients are the substances that support specific metabolic functions. These are vitamins, minerals, and phytochemicals. You can't survive without certain micronutrients, and others while not essential are very healthy (like antioxidants). This is the category that most people mean by "nutrients", and what you seem to be asking about in your question.
Let's take a potato as an example. The potato provides many macronutrients throughout, notably starch and water. A 100g potato has 75g of water, 15g of starch. The flesh contains significant amounts of the minerals phosphorus, vitamin C, and potassium. Now -- ounce for ounce, the potato skin does contain more micronutrients than the flesh. But that's ounce for ounce, and the flesh far outweighs the skin. So in a potato as a whole, the majority of nutrients (both macro- and micro-) are in the flesh.
Which is not to say you should peel them! The real issue is that the skin and the flesh contain different nutrients.
Notice that many fruits and veggies have skins that are more colorful than the flesh - potatoes, apples, pears, peaches, etc. The colors are usually a sign that the skin contains nutritious phytochemicals like carotenoids and flavonoids. Apple skins are red due to the presence of anthocyanins which are not present in the flesh. Anthocyanins are powerful antioxidants. So if you don't eat the apple skin, you don't get the antioxidants. Grape skins contain most of the resveratrol, etc.
On the other hand, oranges and grapefruits (for example) contain plenty of fantastic nutrients in the juice and flesh - orange peel is good for you, but doesn't contain as many nutrients as the flesh. Blueberries are blue all the way through, and the whole darn fruit flesh-and-skin is rich with nutrients. This is similarly true for tomatoes.
In summary: in some fruits and vegetables, the skins contain major amounts of certain nutrients, in particular phytochemicals. But it's wrong to say as a blanket statement for all fruits and vegetables that "most of the nutrients are contained in the peel".