Garlic bulbs need to be able to breathe, so keep them in an open container (away from other foods), in a cool, dark place like the pantry.
Don't store in the refrigerator -- the excess cold leads to sprouting. And don't store in plastic bags because the lack of air leads to high humidity and causes the bulbs to rot.
Unbroken bulbs can be kept up to eight weeks. Once broken from the bulb, individual cloves will last three to ten days.