Helium balloons are any sort of balloon that use helium as their primary gas to keep them aloft. Tethered balloons are what most people think of when they think of helium balloons, which are the smaller party-sized balloons people tie to strings to have float as ornamentation.
Helium balloons float in the air because the gas inside of them is lighter than the air outside. When one gas is lighter than another, it tends to float up on top of it. You can think of balloons floating in the air as being similar to a plastic bottle full of air in a pool of water. Since the air in the bottle is lighter than the water itself, it floats up to the top, even if you put it deep underwater. In the same way, helium balloons will float to the top of the heavier air. The speed at which that bottle of air floats to the top of the water is determined by the difference in their densities, which is pretty extreme: a liter of water weighs 1kg, a liter of air weighs 1g.
Helium balloons don’t float up nearly as strongly as that bottle of air underwater, because the difference in density between helium and air is quite a bit smaller. A liter of helium weighs right around 0.18g, while a liter of air weighs approximately 1.25g, depending on its exact composition. That’s a pretty small absolute difference, so that if we were to fill our same plastic bottle with helium it wouldn’t float in the air, because the bottle itself would weigh more than the difference between the gasses. Helium balloons, however, weigh almost nothing, because they’re very thin bits of plastic. This means that even fairly small balloons will float.
As helium balloons get larger, they are able to carry more and more weight. Since their volume increases at a greater rate than their surface area, if you build balloons big enough, you can actually start to carry people and vehicles. Zeppelins work on this principle, but they have to be quite huge to be able to carry passengers and to make up for the increased weight of the thicker skins and frames they have as their balloons. To give an idea of how the size affects it, an average party balloon with a diameter of one foot (30cm) can carry about 14g of weight. A huge balloon, with a diameter of six feet (1.8m) can carry about seven pounds (3.1kg). And truly gigantic balloons, with a diameter of 100’ (30m) could carry over 16 tons (15,000kg).
Helium balloons are starting to receive some attention at these larger sizes, as possible vehicles for distance air travel. In the past, there was some attention paid to these zeppelins and dirigibles. Unfortunately, some of them relied on hydrogen gas instead of helium, because hydrogen weighs less than half as much as helium, making it more efficient as a gas to float a large amount of weight. The downside, though, was that hydrogen was incredibly flammable. After the hydrogen used resulted in a massive disaster people stopped trusting zeppelins entirely, even though helium is not really flammable at all.