What makes soap float?

Ivory Soap is less dense than water. A bar of ivory soap weighs less than the volume of water it would displace if it were fully submerged. Since the buoyant force is equal to the weight of the displaced water, that produces a net upward force on it. It will rise in the water until enough of it is sticking out of the water so that the buoyant force is equal (and opposite) to the soap bar's weight. When that happens, it floats, just like in the commercials.