To speed up the ripening process in peaches, place them in a loosely sealed brown paper bag on the counter. The peaches give off ethylene gas — a natural hormone present in fruits and vegetables — which is then trapped inside the bag, causing the fruit to ripen faster. You can do this with a plastic bag in a pinch, but plastic doesn't breathe as well as paper, so you run the risk of moisture collecting on the peaches causing them to rot.
Once ripe, peaches should be refrigerated and used within a few days (although they'll taste better at room temperature). Ripe peaches have a sweet fragrance, are soft (but not mushy) to the touch, and have a golden undertone. This ripening trick also works for apricots, nectarines, avocados, bananas, kiwi, mangoes, pears, and plums.