Hard-boiled eggs done right

This should result in an egg that is fairly easy to peel, and that won't have a green rim around the yolk.
Take a thumb-tack or needle and poke a hole into the large end of the uncooked egg, all the way for a thumb-tack, about 1/4 inch for a needle.   Eggs have a tiny air bubble that expands when the egg is heated,and this is what causes the shell to crack.  Pierce the egg (gently, working the thumb-tack slowly in, rotating a bit as you go), and the air has a place to go so the shell won't crack.

Put your eggs in a pot of cold water (six ups of water for up to 4 eggs.  After that, add another cup for every egg you add).  Bring the water to a full boil, remove the pan from the heat and cover it.  Set the timer for 17 to 18 minutes.  At this time, plunge the eggs into cold water. You could have a boil of ice water waiting and move the eggs to that (using a spoon or ladle), or you can quickly drain the pot and run cold water over the eggs, adding ice if needed.

Peel immediately.  Leaving them to sit in warm water is what makes the yolks turn green.
Some people prefer to peel them by lightly tapping all around, so there is a spider web of cracks around the egg. Then you just gently slip the peel off.
I prefer to cut the egg in half, as though for deviled eggs, shell and all, and then use the tip of a spoon to gently work in between the shell and the egg until the half egg comes out of the shell.

If your eggs are very fresh and uncooperative about being peeled, drop 3 at a time back in boiling water for about ten seconds, then plunge into ice water and they should peel.