When vines begin to die back, the tubers have matured. You can leave the potatoes in the ground for a few weeks, but dig them up if you are expecting a heavy frost or having a warm wet spell that would start new foliage sprouting.

Dig carefully with a digging fork or potato hoe, starting from the outside and getting down under the potatoes so that you don't spear or scratch them. Throw away any tubers that show a green discoloration - they are not edible.

Once dug from the hill, allow the potatoes to air dry for about an hour. Store them in a well ventilated lightly humid area with no natural light where the temperature remains cool (40° F). Do not wash the potatoes before storing, and do not pile them more than a foot or so deep.

I agree with Moz8, except that I spread them on paper and allow to air-dry until the surface is dry to the touch - more like a day or two in my area. Always keeps the taters covered (out of any light), to prevent greening. Don't try to store spuds that are cut or nicked.

For home gardeners without cellars or basements: an alternative storage method is to use metal garbage can/s, fill by alternating layers of loose, dry straw and potatoes; lid tightly, and bury in the garden. Put a bale of hay or several bags of dry leaves over the hole (for insulation). The hay/leaves will permit access even in frigid weather, while the soil surrounding the buried can/s maintains proper temperature. Check about every 4 to 6 weeks and remove any that aren't maintaining proper firmness, then re-pack to keep the good ones. I occasionally have a potato go bad, not often but it does happen, probably from getting bruised during the digging process; so handle with care at all stages. Use bruised or cut potatoes right away, don't try to store them.