Floating in General
Most of us feel comfortable with the idea of something floating in water. In fact, people themselves float in water, so we have a way of directly experiencing water flotation. The reason things float in water applies to air as well, so let's start by understanding water flotation.
Let's say that you take a plastic 1-liter soda bottle, empty out the soft drink it contains, put the cap back on it (so you have a sealed bottle full of air), tie a string around it like you would a balloon, and dive down to the bottom of the deep end of a swimming pool with it. Since the bottle is full of air, you can imagine it will have a strong desire to rise to the surface. You can sit on the bottom of the pool with it, holding the string, and it will act just like a helium balloon does in air. If you let go of the string the bottle will quickly rise to the surface of the water.
The reason that this soda bottle "balloon" wants to rise in the water is because water is a fluid and the 1-liter bottle is displacing one liter of that fluid. The bottle and the air in it weigh perhaps an ounce at most (1 liter of air weighs about a gram, and the bottle is very light as well). The liter of water it displaces, however, weights about 1,000 grams (2.2 pounds or so). Because the weight of the bottle and its air is less than the weight of the water it displaces, the bottle floats. This is the law of buoyancy.