I put the eggs into a saucepan, covered with water and a generous dash or 2 of salt. (Just as a side-note: if your eggs crack in the water, you can add about 1 tsp. vinegar to the water and, if they crack, the egg will stay inside the shell and not seep out.) Bring the water to a continuous boil (not just a couple of bubbles, but bubbles over the whole pan).
Then, cover the pan and remove it from the heat. Let the eggs sit for 10-15 minutes or so. I'll then pour out the still-hot water and replace it with cool water (to cool the eggs while I peel them).
Peel them by gently pushing them around on the inside of the sink ~ until the shell is all crackled and soft. Peel them under cool running water (gently push off part of the shell and let the water get underneath it).