Leavening agents

A leaven is anything that produces bubbles in dough or batter, causing baked goods to rise.   Most breads rise because of yeast, which works by fermenting sugar, which in turn produces carbon dioxide.
Baking soda and baking powder are used to leaven quick breads, cookies, and muffins. Baking soda is alkaline and reacts with acid to create carbon dioxide bubbles that become trapped within the batter. It's sometimes used in batters that contain acidic ingredients, like buttermilk or molasses. If there's not enough acid in the batter, the recipe will instead call for baking powder, which combines baking soda with one or more acidic salts. When the baking powder becomes wet or sufficiently hot, the soda reacts with the salts and releases bubbles. Air bubbles can also be trapped in beaten egg whites, a technique used to leaven angel food or sponge cakes. 

* Recipes that call for both baking powder and baking soda are probably using the baking soda to offset extra acidity in the batter (from ingredients like buttermilk or molasses) and to weaken the proteins in the flour.  Omitting the baking soda from these recipes may alter the color or flavor of whatever you're baking, and make it less tender.