New potatoes are best eaten soon after buying them, as they will lose their flavour quickly.
Main crop or "old" potatoes, on the other hand, can be stored in a cool, dark place (to prevent them sprouting or turning green). You should not eat them if they have turned green, as they will not taste good, and may give you stomach ache!
If you buy your potatoes in a polythene bag, then you will need to remove them from it as soon as you get them home. They need air circulating around them if you are hoping to store them for a while. Ideally you would purchase them, unwashed, from a good greengrocer or market stall as they will keep better.