Photo from Jason Doucette's photostream
I keep potatoes all winter and into spring in my root cellar, buried in straw. The standard tips for storing potatoes are storing them out of light, having good air circulation, and optimal temps of 42-55ºF and high humidity so they don't dry out. Potatoes should not be washed before storing, and they should be allowed to cure for 2-3 days in a cool place to harden the fragile skins.
Check carefully for blemishes and soft spots, or damage from digging. Use those right away (or discard) instead of storing. Store in paper bags but not plastic bags so they can breathe. (As I said, I bury mine in straw.) Potatoes exposed to light will develop a green tint which make the potatoes bitter, and eating a lot of green potatoes at one time is potentially toxic.
Potatoes will enter a period of dormancy, and research has shown temperatures of 42-50ºF extends dormancy the longest although that varies somewhat with variety. The colder temps, however, will cause the potato starch to turn to sugar, and those potatoes if fried will be dark. Do not store potatoes next to onions or fruit, which produce hormones and gasses that will spoil both crops.
I always lose a few from rot, probably from a blemish I overlooked when storing them. But, by the time spring rolls around and the temps are warming, the remaining potatoes always start to sprout, even in the absence of light.
Recently I read on The Old Foodie a notion she took from old ship's logs (logged 200 or more years ago). It seems the potatoes stored on ships for the long voyage from England to Australia didn't fare well. One log said if you drop the fresh potatoes in boiling water for a minute or so, then chill immediately, it 'cooks' the eyes just enough that the potato will not use it's energy to sprout, and will keep better on the long journey.
I'm curious to see if it really works, so I think I will experiment with a few potatoes this winter. I always have an abundance from my garden. The only thing is for me to remember that the potatoes I dip have to be set aside so I don't try to use them in spring as seed potatoes!