Not quite. They need an ESTA registration instead of a visa. "Visa Waiver" is the name of the program that allowed citizens of certain countries to visit the US without a visa (usually bilateral) and has been around much longer than the ESTA stuff.
What your ESTA registration acutally replaces is that white I-94 form that you have to fill out while on the airplane. So the border and costums patrol knows that you're not planning to visit the states with criminal intent 72 hours before you're boarding the plane compared to checking that nonsensical form at your destination airport.
I still suspect the whole idea of that is that after years into the visa waiver program, some bean counter noticed that by that program, the US is also waiving billions of visa fees and the found a way to have tourists pay an admission fee even if they don't need a visa. All under the guise of "modernizing" beurocrazy and moving paperwork that used to be ignored for free to a electronic system that earns them $14 per tourist.