Why does a balloon float?

The gas helium is less dense than the nitrogen-oxygen mix of atmospheric air. So a balloon filled to the same pressure (or even higher) will have a lower density than the air around it, which will displace it (flow under it), forcing it upward. The balloon will rise until the combined weight of the helium and the balloon are the same average weight (density) as the surrounding air. Long before it reaches that altitude, a normal balloon will begin to expand and will usually pop (at an altitude of a few hundred to a few thousand feet).
Helium molecules are lighter than most air molecules in the atmosphere (mostly nitrogen and oxygen) therefore they are prone to rise while heavier molecules move downward.