After boiling the eggs -- andeggs—and note that boiling them longer helps to make them easier to peel -- letpeel—let them sit for a while in a pan of cold water. I add ice cubes to the water and put the pan in the refrigerator.
Once cold, crack the "bubble" at the flat end of the egg butby knocking it against the counter or the edge of the sink. Tap a few more times around the rest of the egg to get a few cracks in the shell. Next, gently roll the egg between your palms, applying just enough pressure that you can hear the shell cracking. You want to be distorting the shape of the egg enough that the now-solid part in the middle flexes away from the skin and shell.
Finally, starting at the cracked "bubble""bubble," work on separating the skin beneath the shell. The idea is not to peel the shell, per se, but to peel the skinskin, which will also slough the shell.
The only challenge arises when the skin won't separate from the solid inner part. Once you're down to picking bits of the shell off, you've lost. It's then very difficult to maintain the integrity of the egg's surface.
I haven't found running water over the egg to be helpful. It makes the shell sticky. Better is to cook the eggs a little longer and soak them afterward, so that there's still some moisture sitting beneath the skin. If the skin remains moist (and, hence, thicker), it will separate more easily from the solid inner part of the egg.
Always boil a few more eggs than you'll need, and, if the intention was to present the eggs as, say, deviled eggs, use the rejects for egg salad.