Baking soda and bicarbonate of soda are actually different names for the same thing, but here in Australia it is mainly referred to as bicarbonate of soda.
Both bicarbonate of soda and baking powder are leavening agents. When included in a batter, the leavening agent creates air bubbles that expand when cooked, and cause it to rise.
Bicarbonate of soda needs to be mixed with moisture and an acidic ingredient for the necessary chemical reaction to make it rise, such as lemon juice, chocolate, buttermilk or honey.
Baking powder often comes pre-mixed with the acidic ingredient for you – so all you need to do is add the moisture. To make your own baking powder, mix two parts cream of tartar with one part bicarbonate of soda.