The answer is one word: buoyancy. What does that mean? Well, let's think of an easy example: a piece of wood floating on water. What makes the piece of wood float? It turns out that when you put a piece of wood in water, some of the water has to move out of the way to make room for the wood; this is called displacement. When this happens, the water responds by pushing on the thing causing it to displace (this is because water has pressure). This upwards push is called the buoyant force. If the object displacing the water is light enough (the proper word to use in physics is "density"), the water can push up on it hard enough to balance the force of gravity pulling it down, and so it floats. Since wood is less dense than water, it floats!

So, how is this relevant to a hydrogen balloon? Well, it's exactly the same thing. You can think of the atmosphere (the air around us) as the water, and the balloon as the piece of wood. The balloon has to displace some air, so in response, the air pushes up on it. Now, most things are much denser than air, so we barely notice this effect. But hydrogen is less dense than air, so the air pushes up on the balloon harder than gravity is pulling it down, and the balloon tries to move up! So just like wood float on water, hydrogen balloons float on air.

Objects float when immersed in fluids due to differences in their densities. Less dense objects float in more dense materials. Density is a measure of how much substance per unit volume an object has. For example, styrofoam (think about how porous it is!) is less dense than water, so it floats on water.

So why do hydrogen balloons float in air? Air is a fluid that is comprised of about 79% nitrogen, 20% oxygen, and trace amounts of other gases. Since most of air is made of nitrogen and oxygen, which have heavier molecular weights than hydrogen, air is denser than the hydrogen-filled balloon (the atmosphere is denser than the hydrogen-filled balloon). This results in the balloon floating in the air!