Very fresh eggs can be difficult to peel. To ensure easily peeled eggs, buy and refrigerate them a week to 10 days in advance of cooking. This brief "breather" allows the eggs time to take in air, which helps separate the membranes from the shell.
To peel a hard-cooked egg, gently tap egg on countertop until shell is finely crackled all over. Roll egg between hands to loosen shell. Starting peeling at large end, holding egg under cold running water to help ease the shell off.