Peaches are best kept at room temperature but are also not harmed by refrigeration. To make your peaches last all week, eat the softest first, place firm ones in the bottom of the fridge, and rotate "a day's worth of peaches" from the fridge to the counter each day you eat peaches. This way, tomorrow's fruit will ripen as you eat today, and the rest will be waiting for you in the refrigerator. Though it varies with peach varieties and weather conditions, peaches commonly last one week or longer in a refrigerator.
Can't Wait Any Longer? Fruit ripens in the presence of a gas called ethylene, found naturally in the fruit. A cool trick to quickly ripen your "firm ripe" peaches should you find yourself in need is to cut a piece of a high ethylene fruit; an apple or banana, and place it in a plastic bag with the peaches you wish to ripen, tie the bag shut at the top and place on the counter or in the fridge (depending on how dire the peach emergency is - don't panic, just place bag on counter). It won't take long at all using this method.