leavens = 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 = bicarbonate of soda = sodium bicarbonate

Baking soda is alkaline, and when mixed with acidic ingredients, it reacts and releases bubbles of carbon dioxide.  These bubbles, when trapped inside batter, help baked goods rise.

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.

Baking powder   

Baking powder is a mixture of one or more acidic salts and baking soda, an alkali. When the baking powder becomes wet or sufficiently hot, the soda reacts with the salts and release carbon dioxide gas bubbles.

Most recipes that call for baking powder intend for you to use double-acting baking powder.  This includes two acidic salts--one that reacts when wet and one that reacts heated.  By giving the baking soda two chances to react, it usually results in light and airy baked goods.  Less common is single-acting baking powder, which only reacts when it becomes wet. When using this kind of baking powder, you have to get the batter into a preheated oven immediately after you mix the wet and dry ingredients together. 

Yeast 

Yeast is a one-called fungus that converts sugar and starch into carbon dioxide bubbles and alcohol.   This has made it a useful ally in the production of bread, beer, and wine. There are many varieties of yeast.  Bread is made with baker's yeast, which creates lots of bubbles that become trapped in the dough, making the bread rise so it's light and airy when baked.