That somewhat depends what that date is on the carton - a pack date or expiration.
If it is a pack date, they should be good for a month if stored properly. If it is an expiration, they will still be good to eat if the eggs have been properly handled and stored under refrigeration. See Related Links.
As for how long they will be good, the older they are, the whites become thin and the yolks won't stand so tall.
Some suggest placing the eggs in water. If they float, consider them not good, although a spoiled egg generally leaves no doubt in your mind once you break the shell. That's one reason why cooks learn to crack eggs separately to check them before using them in recipes.