Anionic Surfactant
Surfactants are sorted by what their water-loving end does about charge; the anionics carry a negative one. Soap is the oldest example, the sulfates are the best known, and the milder modern ones — the glutamates, sarcosinates, taurates and isethionates — belong to the same class despite being sold as the alternative to it. Sulfate-free does not mean surfactant-free, and it usually does not mean anionic-free either.
The charge is why they clean well and why they are the class most associated with irritation. Anionics interact with the proteins of the outermost skin layer as well as with oil, and that interaction — swelling, unfolding, and easier passage for whatever else is in the formula — is much of what a harsh cleanser is doing. Dermatology research leans on it: sodium lauryl sulfate is a standard way to provoke controlled barrier disruption in study skin.
Two things are worth separating from harshness. Foam is a class property rather than a strength reading, and a very mild anionic can lather generously. And pH is not set by charge alone: soap has to be alkaline to exist at all, while a synthetic anionic can be formulated near the skin's own pH, which is one reason a modern gel and a bar feel so different.
Position on a label matters too. High in a cleanser an anionic is the product; far down another list it is usually solubilising a fragrance rather than washing anything.