The fresh, young sprouts of many plants are tender and tasty, and in some cases more palatable than the mature plant. Unfortunately, in the case of potatoes, sprouts are distinctly undesirable. The humble, familiar spud is defended by some highly toxic alkaloids, which are concentrated in their eyes and in those juicy-looking new shoots.
The existence of sprouts on your potatoes is more than a cosmetic change. It means that the root is trying to fulfil its natural purpose of growing a new potato, and is beginning to convert its stores of starch into sugars to fuel the new plant. More importantly, it protects the infant plant from predators by filling those sprouts and their underlying eyes with a blend of toxic alkaloids. As long as you diligently remove the sprouts and the "eyes" where they're attached, where the potentially harmful alkaloids are concentrated, you can still safely eat the potatoes.
If the sprouts are accompanied by a visible green tint in the underlying potato, its safety is more problematic. That green color means the potato has been exposed to lots of light, and it signals the toxic alkaloids -- most notably one called solanine -- are widely distributed throughout the potato's skin and underlying flesh. Solanine can cause serious illness or even death, when consumed incautiously, and its levels vary widely from potato to potato.
The Paring of the Green
To salvage a green-tinged potato, you'll need to perform ruthless surgery. Cut away the sprouts, the eyes, and every part of the potato that contains a trace of green. Solanine and its related alkaloids are most concentrated just beneath the skin, so paring it to a depth of at least 1/4 inch should render the potatoes safe to eat. The alkaloids are noticeably bitter in flavor, so if you find your potatoes are bitter even after peeling you should not eat them.
When you find yourself with a large bag of dubious tubers on your hands, exercise some judgement. Potatoes with large moldy or rotten spots are best discarded, because such a small portion of each spud will remain usable. If the potato is small enough or flat enough that cutting away its green areas isn't practical, discard that as well. Any that feel sound, and are large enough to safely trim away their eyes and green areas, can still be prepared and eaten.
The Ounce of Prevention
Storing your potatoes properly will help prolong their shelf life, and minimize sprouting. For long-term storage they need a cool, dry place, such as a cellar, where the temperature will not vary greatly. Warmth and humidity will both encourage the potatoes to grow, and should be avoided as much as possible. The potatoes also need good airflow, because still air allows moisture to accumulate and promote mold and mildew. Finally, keep your potatoes in the dark. Light encourages the growth of sprouts and the production of solanine.