How can I cook the perfect potato for a potato salad?

First, potato is not the same as potato. Conventionally, potato salad is always made with waxy potatoes, because mealy potatoes' outer layers disintegrate when tossed with the sauce, much like making risotto. They are also less creamy in texture. But there is also the school of using mealy potatoes, because they absorb more seasoning, and also because some people like the soft texture. If you decide to go with mealy, use an acid in the cooking water (depending on your seasoning, choose vinegar, citric acid, or the neutral tasting cream of tartar). This will firm the potatoes. It isn't necessary for waxy potatoes.

Second, the cooking time depends strongly on size. You must cut them very evenly. Else you will have both undercooked and overcooked potatoes in the pot at any given moment (up to the time when you only have overcooked ones). Smaller pieces are easier to work with.

Third, you want to heat them as gently as possible. Start them in cold water, and cook them at a simmer, not at a boil.

There is no way to predict when the potatoes will be ready. Even if you cut them to the same size every time, the sort and the age of the potatoes will result in varying cooking times. When the time is near, you must look after them and try them constantly. Take out a piece, cut it in half, cool it in cold water, and chomp on the exact center. If it isn't crunchy, stop the cooking immediately and remove the water. The time window in which they are just right is quite short. Season them while they are hot, they will then absorb the sauce instead of just swimming in it the way cold potatoes do.

There are many different varieties of potato salad.

I've had an excellent one that was actually made from mashed potatoes. (starchy potatoes cooked, mashed, then the other ingredients mixed in; there was celery and other vegetables to add variety so it wasn't all mush)

America's Test Kitchen has an excellent recipe for an Austrian-style potato salad where they boiled medium-starch potates in chicken stock spiked with vinegar, then mash some of the potates to thicken the liquid which was reduced to use as the dressing. They also had a science segment where they described how the vinegar gives you a longer window for the 'cooked but not mushy' stage, but that it also increases the cooking time.

But rumtscho's correct, in that the typical 'potato salad' potato uses waxy potatoes, which will not turn mushy as quickly when cooked. I typically use two different methods for testing doneness, though -- (1) a knife inserted into the middle of the potato of an undercooked potato will get stuck. The potato will fall off under its own weight when it's cooked. (2) if cut open, an undercooked potato will be a different shade in the middle than on the edges ... there's a pretty distinctive delineation, too, so you can just look to see if it's of one tone, or two tones.

It's also useful to simmer, rather than actually boiling potatoes.

The Food Lab did some experimenting with potato salad and found that you should..

• Season the potatoes while they're hot (or season the water they cook in). The seasonings won't absorb once the potatoes cool.
• Add vinegar to the water that the potatoes are cooking in. This will prevent them from breaking apart.
• Use Russet potatoes. They absorb seasonings better.