Many volcanic rocks are not light at all. But certain ones, especially those that come from magma rich in water are low density because they are samples of frozen foam! That is, when the magma gets close to the Earth's surface, the pressure is low enough so that H₂O initially dissolved in the molten stuff comes out of solution. This is analogous to what happens when you pop open a bottle of soda! In this case the CO₂ that is dissolved in the liquid comes bubbling out. Now, in the case of magma, since it solidifies at 900 degree C, the bubbles become frozen when the temp drops and we have volcanic foam. Some volcanic rocks that are very vesicular (called PUMICE) actually float on tap water.

Great question. You probably remember that volcanic rock forms when a volcano erupts, spewing hot lava and ashes everywhere. When that lava and ash cools down, it turns into solid rock. But -- this is the key point -- lava and ash often has a lot of air mixed into it during the eruption, resulting in pockets of air trapped in the cooled-down rock. It's those air pockets which make many volcanic rocks lighter than other rocks. (But the amount of air trapped can vary a lot among volcanic rocks, so some volcanic rocks can be quite dense.) If there's enough air trapped, the volcanic rock *could* float on water -- but the vast majority will sink, just like any other rock. Hope this helps!

Not all volcanic rock is 'light' or low density -- basalt is relatively heavy and is organic, granite is lighter, but still volcanic in origin. I think you are thinking of gabro and volcanic bombs -- these are rocks with dissolved gasses which caused bubbles to form when the pressure was released (by the eruption). Due to the bubbles, these rock can indeed be very light and many will float on water.