Wait until late in the season and take only what you need, carefully resetting the plant so the smaller tubers have a chance to mature. […] For winter storage, it’s best to let the plant and the weather tell you when to harvest potatoes. Wait until the tops of the vines have died before you begin harvesting. Potatoes are tubers and you want your plant to store as much of that flavorful starch as possible. (gardeningknowhow.com)

If you prefer to store your taters, you should wait 2-4 weeks after the plants have died back to harvest, to toughen them up a bit. When you see that the plants have mostly died back, stop watering altogether to prevent rot or re-sprouting of your crop. Since the potatoes are now finished growing, there is no longer any need for water and leaving them in-ground for a couple of weeks extra allows the tubers to develop a thicker, stronger skin, which is needed for safe storage to protect them from rot or disease. If you have a very late crop, you can start the process and harden them off before the plants die back by cutting down the stems and foliage and stop watering. (countrybasket)

If you want to store your potatoes, harden them off in-ground first, as described above. After digging, let them cure for several days to two weeks by letting them dry on newspaper or dry ground in a dark space, such as the garage, basement, etc. This hardens the skin further for better storage. Make sure to keep the spuds away from sunlight, and do not allow rain or any water to reach them. Do NOT wash them if they are to be stored. Excess soil can be gently brushed or rubbed off, but make sure not to damage or rub off any of the skin. Damaged potatoes should not be stored, but used within a few days, although minor cuts and bruises will heal during the curing period. (countrybasket)

Potatoes should be stored in a cool, but not cold, dry, dark environment, ideally around 45-50 degrees F (7-10 degrees C). Do not refrigerate them as they will turn sweet and change consistency, though this can be reverted by bringing them into warmer temperatures for a day or two. Potatoes do not tolerate frost. (countrybasket)

Potatoes can be stored in bins, baskets, cardboard boxes, etc., or even in paper sacks or mesh bags. They do need good ventilation, so the container should be well perforated. Never store in air-tight containers. It is best not to pack them too tight, but keep some air flow around them to prevent rotting. Make sure they are kept in a very dark area, or cover the bins with newspaper or cardboard to keep all light out, to prevent them from turning green. (countrybasket)