Potatoes, which are tuberous members of the Solanaceae, or nightshade, family, commonly begin growing new sprouts from their "eyes" when appropriately stored, but they are completely safe to eat. In fact, it is this same characteristic that allows one potato to begin growing a new plant.
However, because potatoes belong to a family of plants known for a highly toxic alkaloid, there are some things people need to watch out for. For example, green patches of skin form when potatoes are exposed to direct sunlight for long periods, and even fluorescent lights can cause this to happen. Avoid eating the green patches of the potato because this flesh will contain higher concentrations of solanine, which can be lethal. This is why potatoes have traditionally been stored in a cellar or other part of the home that is cool and dry. However, even with proper storage, potatoes will begin to soften. This is still safe to consume, but watch for extremely soft patches or areas of the skin that puncture easily with a finger because this is a sure sign that it is beyond the point of safe use. Potatoes, like most tubers, store for long periods in the right conditions, so chances are high that even last year's potato crop is still edible.