Place the eggs in the bottom of a saucepan and cover with cold water by at least an inch and add the salt (about 1 teaspoon salt for every 6-8 eggs). Heat the eggs uncovered over medium-high heat until the water comes to a rolling boil.
Immediately take the pan off the heat. Cover and let sit for 15-17 minutes (the exact time will depend on how firm you like the egg yolk and the size of your eggs). I let large eggs sit for 16 minutes on the dot and they are perfect.
While the eggs boil, make an ice water bath in a large bowl. When the eggs are done, scoop them out one by one with a slotted spoon and place them in the ice water bath to shock the eggs and shrink the membranes (discard the hot water in the pot). Leave them in the ice water bath for a few minutes. They can be stored in the refrigerator at this point or cracked and peeled.
A few tips on peeling: crack the egg all over and start peeling at the base (the wider end) of the egg, getting your fingers underneath the membrane of the egg. Peeling them under a stream of slightly warm water seems to help too.
Notes
Eggs: a key component in easy-to-peel hard-boiled eggs is the age of the eggs. The older the better. So if you plan on boiling eggs, buy them in the store and keep them in your fridge for a week before boiling. It will make a huge difference - fresh eggs are a pain to peel.