Potatoes grow easily from seed potatoes. You've probably unintentionally started a few of your own just by keeping a bag of potatoes around long enough for them to sprout. The potato is a tuber, which contains all the material necessary to start new plants. The tubers grow underground from the stem, not from the roots.
Look for potatoes that already have dimples. That's where the eye of the potato will grow. The eye is actually the beginnings of a new plant. Grocery stores often have several different kinds of potatoes, including red, fingerlings, russet, purple and Yukon gold. Potatoes in stores are treated to resist sprouting. Either buy organic potatoes or buy more potatoes than you think you'll need for sprouting.
Place potatoes where they will receive light but not where it's hot and dry. The potatoes should send up sprouts in a week or two. One trick is to place the potatoes in cardboard egg cartons so they don't roll around.
Cut the potatoes so that each piece has one or two sprouts and a good amount of potato, about 1 1/2 inches square for each seed.
Let the cuts heal over into a callus, which should only take a day or two. Your seed potatoes are now ready to plant.