Short answer: in a dark and cool (50- to 60-degree) place. The refrigerator is also fine. Our tests have determined that potatoes should last about three to four weeks under these conditions with very little shriveling, discoloration, or sprouting.
We also tested storing potatoes on a sunny countertop, in a warm storage area, and in a drawer with onions. These all turned a bit green over the course of a few days, an indication of the development of solanine, a toxic compound. We had to toss them.
If you want to get really serious, turn to an old trick from Maine and store them with an apple. The ethylene gas released by the apple, along with some other airborne compounds, will inhibit sprouting and keep your potatoes usable for up to two months.
Then again, potatoes never stick around for two months at my place. I like eating them too much.