When water is boiling, little pockets of water everywhere (especially near the heat source) are being vaporized into steam: water is turned from liquid phase to gas phase. This is different from bubbles in soda, which are made of carbon dioxide that was dissolved in the liquid. It is also different from evaporation, which usually doesn't involve bubbles and happens right at the surface of the water.

Let's categorize all of these terms. In general, the phase change from liquid to gas is called vaporization, and it can happen in two ways. The first way is evaporation, where liquid molecules on the surface vaporize and escape into the air. Evaporation does not require high temperatures. Vaporization can also happen through boiling, especially nucleate boiling, where liquid molecules vaporize because they are heated above the boiling point temperature. If the heat is coming from a surface at the bottom of the liquid, like a flame under a boiling pot, that's probably where the gas is being formed. Since this gas is inside a liquid, it packs itself into the shape of a bubble and rises through the liquid until it escapes.