Baking powder is baking soda with the acid built right in (usually in the form of cream of tartar, a by-product of the tartaric acid produced in winemaking). The two active ingredients are usually suspended in an inert medium like cornstarch. The reaction takes place as soon as the compound is wet, and proceeds briskly, so the rising can happen right in the oven (and will happen right in the bowl if you don’t move fast).
Double-acting baking powder gives an initial release of carbon dioxide that bubbles up and makes a batter expand within minutes of being added, followed by a second rising in the oven.