Put the raw eggs carefully into a pan large enough for them to roll around a little. Cover them with water and bring to a boil over medium heat. Don’t rush this or the eggs will crack.
Bring the eggs to a rolling boil, remove from the heat and set aside for ten minutes. (If you live at a high elevation, like several thousand feet, let them boil for three or four minutes extra before taking them off the heat.)
Drain the water from the pan and fill with cold water to stop the cooking. Then drain again and shake the pan around to bang the eggs together to crack the shells. Add cold water, with ice, if you’ve got it. As soon as the eggs are cool enough to handle, peel them under running water and set aside. Start peeling at the large end. There’s a thin membrane between the shell and the egg; if you get under the membrane it helps separate the shell from the egg.
If you are making hard-boiled eggs for future use, I still recommend peeling them now, and bagging the peeled eggs in the refrigerator. Later they will be much harder to peel.
Cut the peeled eggs in half lengthwise and scoop out the yolk into a small bowl. Mash with a fork, and add mayonnaise, Dijon mustard, salt, pepper, and, if you wish, sweet pickle relish or chopped sweet pickles. You’ll figure out how much to use by tasting it, but in general you will need about 1/4 cup of mayo and 1 tablespoon of mustard for six eggs.
Spoon the mixture back into the egg whites, and garnish with paprika.
If you want to experiment, add some of your other favorites to the mixture, such as chopped jalapeño, green chile, or onion. I’ve also seen recipes that add chopped almonds. Heck, make your own egg recipe and name it after yourself.