What a great question, and quite a tricky one to answer. You probably know that if something starts moving, then there must be a force on it: that’s one of Newton’s Laws.
But where does the upward force on the balloon come from?
It comes from the air.
You probably know that the air pressure is less at the top of a mountain than it is at sea level. This is the basis on which altimeters work.
But perhaps you didn’t realise that the air pressure is slightly less at the top of a balloon compared to the bottom of the balloon. In fact the air pressure falls by 0.011% for each metre increase in height.
The change in air pressure with height gives rise to a force on the bottom of the balloon pushing it upward that is just a little bit larger than the force on the top of the balloon pushing it downwards. We call this force buoyancy, and it affects the weight of everything when it is weighed in air.
So for example, for precision weighing, the biggest correction is estimating the magnitude of this buoyancy. For a normal balloon, the buoyancy reduces its effective weight. For a balloon filled with a gas that is less dense than air, the buoyancy is enough to make the balloon rise.
How long can they last?
Helium balloons are temperature sensitive and if they are not totally full, so there is room for the helium to expand, then they can last for days, or even weeks.