Why does a helium balloon rise and a lead ball sink in air?

Both are doing the same thing. Air is a fluid, and anything in it is pushed up by a force equal to the weight of the air it displaces. If the object weighs less than that air, it goes up; if more, it goes down. The balloon is not being pulled upward by anything - it is being outcompeted by the air trying to get underneath it.

That means everything is slightly buoyant in air, including you: you are being held up by about a newton, and a precise weighing has to correct for it. Helium is about a seventh the density of air, hydrogen half that again - which is why airships used hydrogen despite the obvious problem.

It also explains why a balloon stops rising. Air thins with altitude, so eventually the displaced air weighs no more than the balloon; a weather balloon expands as the outside pressure drops until the skin bursts, typically around 30 kilometres. Archimedes worked the principle out in the third century BC, in water, and the story about him leaping from the bath comes from a Roman writer two centuries later.