To make successful hard-boiled eggs that peel easily, place eggs and water into a saucepan, add 2 tablespoons of vinegar and 1/2 teaspoon of salt. Boil rapidly for 5 minutes, then allow to sit for 15 minutes. Move eggs to a bowl of icy water for 5 minutes before peeling.
Adding vinegar to the water enables the eggs to have whiter whites and brighter yellow yolks. It also helps the eggs to peel easier. It is important to cook the eggs in enough water that they are fully submerged when placed into the saucepan. The 1/2 teaspoon of salt adds flavor.
Another thing to note is that fresher eggs often are more difficult to peel. Use eggs that have been in the refrigerator for a while -- sometimes as long as 10or more days.
After cooking and moving the eggs to an ice bath, be sure to allow the eggs to rest for a few minutes outside of the ice bath before peeling. Eggs closer to room temperature are easier to peel than eggs that are still very cold.
If the hard-boiled eggs will be stuffed later, it is best to let them soak in the ice bath with the pointed side facing down so the yolk shifts more to the center of the egg.