"Peel", "pare", and "skin"

The meaning of those verbs is:

• pare: trim something by cutting away its outer edges; cut off the skin of something
• peel: remove the outer covering or skin from a fruit, vegetable, or shrimp
• skin: remove the skin from an animal, a fruit, or vegetable

Carlo pared his thumbnails with his knife.
Peel off the skins and thickly slice the potatoes.
He scrambled down from the tree with such haste that he skinned his knees.

_{[Reference: the New Oxford American Dictionary.]}

While peel and skin are both nouns and verbs, pare is only a verb, from which I deduce you're talking about these words used as verbs. So:

• peel is primarily used for fruits and vegetables (peel an apple), as well as clothing (in the sense of removing: to peel off one’s pullover); it can also refer to other coverings on the surface of things
• skin is mainly used for persons, animals (skin the cat!) and fruits
• pare is, in my book, used much less frequently, but is as generic as skin

So, as I see it, skin is the most generic term, while peel refers mostly to fruits and vegetables (you probably wouldn't say you're peeling the cat). pare is much less used, but as broad as skin.

ldoce defines:

• Peel [noun]: the skin of some fruits and vegetables, especially the thick skin of fruits such as oranges, which you do not eat:

orange peel

• Peel [verb]: to remove the skin from fruit or vegetables

Peel and dice the potatoes.

• Skin [noun]: the natural outer cover of some fruits and vegetables, the word also has many other meanings.

banana skins

• Skin [verb]: to remove the skin from an animal, fruit, or vegetable

Add the tomatoes, skinned and sliced.

• Pare [verb]: to cut off the outer layer of something, using a sharp knife

Pare the rind from the fruit.

It's good to know:

• rind [noun]: the thick outer skin of some types of fruit, such as oranges

grated lemon rind

• zest [noun]: the outer skin of an orange or lemon , used in cooking

grated orange zest

"pare" has additional meaning and use over "skin" and "peel" in that "skin" and "peel" refer to the removal of a thin outer layer of something (as indicated in the other answers).

"pare" is often used to indicate something was reduced in size or number, but in small increments. It is quite common to hear someone speak of paring something down to its essential elements or paring back the budget for the next fiscal year.

One might think of this as peeling something, then peeling it again, repeatedly.