The key to easily peeling a boiled egg is cooling the eggs before attempting to peel them. This contracts the egg inside the shell, allowing a larger air bubble to form and resulting in an egg that is much easier to peel.
After boiling the eggs, drain your saucepan of its hot water and refill it with cold water. Allow your eggs to soak in the cold water for a few minutes. After this, take your eggs out of the saucepan and lightly tap the wide end of an egg against a hard surface, cracking it. Using your thumb, peel away the shell and the white, film-like membrane to reveal the smooth, shiny egg. If this doesn't work, try using a spoon to scoop out the egg after the initial shell-cracking.