The natural inclination is to pull a grape off a cluster and pop it straight in your mouth, skin and all. The skins are edible, but may prove difficult to digest, particularly for small children. Peeling a grape makes digestion easier and eliminates the risk of illness from pesticide residue. The grape skins also need to be removed when you make grape jelly and jam, because the skins don't break down with the pulp.
Grape skins are similar to tomato skins -- the flesh inside is very soft and juicy so you want to avoid bursting the grapes as you peel. Peel individual, raw grapes with a paring knife just as you would peel a raw tomato or apple. Start at the stem end where you find a slightly raised circle, using the blade to lift up and dig under the raised skin. At this point, you can usually hold your thumb over that bit of skin and simply pull down to remove most of the skin, repeating over the entire grape. If the skin breaks off, use the edge of the knife to lift up the edge and continue peeling.
A batch of grape jam or jelly might require hundreds of grapes, which would be tedious and time consuming to peel individually when raw. The grapes are going to be cooked down to make the jam or jelly anyway, so save yourself time by cooking them before peeling. Bring a pot of water to a boil, and drop the grapes in for at least 10 seconds or until you notice the skin splitting if you simply want whole, easy-to-peel grapes. Grapes for jelly are typically smashed and simmered for about 10 minutes. Plunge whole grapes into ice water to cool them, and simply pull off the hanging skin with your hands. For jelly purposes, add the cooked grapes to a food mill. Turn the crank to push the pulp and juice through while leaving the skins behind. Strain the pulp through cheesecloth to remove finer pieces.