Baking powder is composed primarily of baking soda and an acid, phosphate salt (such as cream of tartar), with which it reacts to produce carbon dioxide. Carbon dioxide expands the batter.
Double-acting refers to a process that occurs in two stages. Some release of the carbon dioxide (gas) occurs when the powder is added to the cold batter. But most of its leavening power is released during baking. (When I began baking, single-acting baking powder was also available. Its full leavening power was released immediately.)