The easiest way to keep potatoes white after peeling or cutting is to fully submerge them in a bowl of water until use. If they are not needed until the next day, they can be refrigerated in the bowl of water overnight.
Though the discoloration can be unappealing, potatoes whose cut sides have become brown are still safe to cook and eat. The color change is caused by an enzyme and iron-containing phenols in the tuber reacting with oxygen. Just as iron oxidation turns an old truck rusty-brown, the same process occurs with produce. Covering the potatoes with water halts this reaction by removing the exposed flesh from oxygen.