Mature peaches will continue to ripen after they are picked,* so when it comes time to store them, you have several options. In all cases, if they can't be on a tree, peaches like best to sit on their shoulders and not touch one another as shown. If, however, indulging their solitary nature is too inconvenient for you, go ahead and put them in bags or bowls together, just try to avoid stacking too many on top of any others. Once you have that in mind, decide how to store them based on how ripe they are.
Perfectly Ripe Peaches
If the peaches you have at hand are perfectly ripe but you aren't quite ready to eat them, simply store them in the fridge. The cold will radically slow down their off-tree ripening talents. Check chilled peaches frequently; the cold air in the refrigerator is dehydrating, so watch out for any wrinkly skin, a sign of both drying and over-ripening. (In short, a perfectly ripe peach should be eaten at the earliest convenience.)
If, however, your peaches could stand to be a wee bit softer, a wad more fragrant, keep them on the kitchen counter. You can even let them sit in a spot of sunshine to hasten things a bit (just make sure that spot doesn't get too hot).
For peaches that are still quite firm and you'd like to get them moving along the ripening path, put them in a paper bag—it will capture the ethylene gas they give off naturally and speed up the timeline of their perfection.
Speed Things Up
Still not quick enough for you? Add a banana—the riper the better—or an apple or pear in the bag. These fruits give off even more ethylene than peaches and will coax the peaches along.
When you've kept peaches on the counter or used the paper bag method, once any given peach is ripe be sure to eat it or move it to the fridge until you're ready to eat it so it doesn't go from ripe to rotten.
Let Them Breathe
In all cases, peaches need to breathe, so paper bags or plastic bags with holes cut in them are your best bet if you want to enclose them in some fashion. Just remember, the paper bag will push along their ripening unless you put it in the fridge.
For Longer Storage
Can't eat all the peaches at hand quickly enough? Freeze them! Frozen peaches are fabulous in baked goods and smoothies and for ice cream. See How to Freeze Peaches for more details.
*Please note that the subject is mature peaches. Despite peaches' superlative ripening talents, peaches that are picked too green, this is before the fruit is fully mature, will never ripen properly. This is definitely a situation where one has to work with nature, not try to trump it!