Potato salad can be made many different ways and with many different kinds of potatoes, but the potatoes are always (or almost always) boiled. You want them to be perfect—neither too hard nor too soft—before you add them to your preferred recipe.
Peel the potatoes if you are using thick-skinned ones. Thinner-skinned red and white potatoes can keep their skins if you prefer to have the extra healthy nutrition and color they add to the potato salad.
Cut the potatoes into halves or fourths lengthwise so they will cook evenly and quickly. Place in the pot with just enough water to cover them. Add about 1/2 teaspoon to 1 tablespoon of salt to the water, depending on how many potatoes you're cooking.
Bring potatoes to a boil on medium-high heat. The water should boil enough to bubble fully, but watch carefully as the foam on the surface of the water can make it boil over onto the stove. When the water begins to boil turn the heat down enough to keep a good bubbling boil without boiling over.
Test potatoes with a fork after 8 minutes to see if they are soft enough for the fork to enter easily. If not, recheck every couple of minutes. Do this until they are done--but not overdone! Potatoes that are overcooked and change the consistency of the potato salad.
Remove pot from heat. Turn off the stove burner and pour off the boiling water. Drain off the water with a colander and then place in a large bowl or pot of cold water. (If you cool them at room temperature, they'll continue to cook a bit beyond the stage they were when you removed them from the heat.) Once your potatoes they are cooled, they're ready to add to the rest of your ingredients.