We were going to start with our standard line - "there are many schools of thought on this question" - but it turns out while we weren't looking that most of the cooking world seems to have reached agreement.
The first point to be made, though, is that you don't want hard-boiled eggs, you want eggs that have been simmered gently. Otherwise, the whites become tough and rubbery, the yolks pale and chalky, and the egg is quite likely to develop a gray-green line between the yolk and white.
The first step is discretionary - you can prick a hole in the large end of the shell with a clean thumb tack, pin, or egg pricker, which should help keep the shell from cracking as the air in the egg expands. Next, put the eggs in a single layer in a pot. Add enough water to cover all the eggs by an inch. Turn on the burner and bring the water just to a boil. Remove the pan from the heat, cover, and start timing. The eggs should cook for XX minutes.
OK, we lied. Not everyone in the food world is in complete agreement on the amount of time you let the eggs cook. Julia Child, in The Way to Cook, says 17 minutes "exactly." David Joachim, in the modestly titled Brilliant Food Tips and Cooking Tricks, says 15 minutes produces "perfect hard-cooked eggs." Rosemary Brown's Big Kitchen Instruction Book takes you by the hand and tells you medium eggs need 12 minutes, large eggs take 13, extra-large eggs take 14, and jumbo eggs require 15 minutes.
Julia Child tells all cooking-wise
Adding just a little more dissent, the worthy committee that wrote a recent Joy of Cooking says you can add your eggs to already simmering water, keep it simmering, and cook the eggs for 12 to 15 minutes, taking into consideration the size of the eggs and whether they just came from the refrigerator or not. If the eggs were just taken from the refrigerator, they will take a couple minutes longer than if they went into the drink at room temperature.
Everyone agrees you should immediately place the eggs into cold water to stop the cooking once the time is up.