If eggs are never refrigerated, they last quite a while (weeks!) at room temperature. Once they are refrigerated once, though, they MUST be kept in the fridge or they go bad quickly.
That said, they will last the longest if refrigerated.
At our house, we go through eggs so fast that they're never sitting around longer than 5 or 6 days. We keep them in an egg basket, so they get turned over all the time when taking out/adding eggs.
if you plan to keep eggs for several weeks, especially if at room temperature, you should turn them end-for-end every few days (just trun the egg carton upside down). This prevents the yolk from settling against the membrane, as I understand it, and spoiling quicker. We kept fresh unrefrigerated eggs for many weeks like this when we lived on our sailboat in the tropics. Just turned them over every few days. When they got to be over 3 weeks old we would test them by dropping in a glass of water. If they float to the top they are BAD, toss them. If they sink to the bottom they are fresh. Bob in the middle, they're old but not bad yet.
Stacey