Why do helium-filled balloons float?

Helium is the second element on the periodic table. It is the second because it is the second smallest element by atomic weight, after hydrogen. 

The air we breath is made up mostly of nitrogen (80%) and oxygen (20%).  Both of these elements come later on the periodic table (7th & 8th) because they have a higher atomic weight than helium. 

When you fill a balloon up with helium, you are filling it with air that is lighter than the air surrounding it.  Therefore, the balloon floats.  In the same way, oil is less dense (lighter) than water, so it will also float when surrounded by water.    

If you were to fill a balloon with just an air pump using the air around you, it would not float since the air in the balloon is equal (not lighter) to the air surrounding it.