Balloons that have been inflated by air fall to the ground because the density of the gas inside the balloon is greater than the density of the surrounding air. Air contains a mixture of nitrogen, oxygen and trace gases, while the air in balloons is mostly carbon dioxide.
The molecules that make up carbon dioxide are more closely packed, which makes it a denser gas than either nitrogen or oxygen. The balloon sinks because the air is unable to support this denser gas. When a balloon is filled with a gas that's less dense than the surrounding air, such as helium, the balloon floats.