Think of your respiratory system as an upside-down tree. When air enters your body, it goes down your trachea. That's the trunk. Then it quickly divides into two airways (the tree's limbs). Those are your bronchial tubes.
Then, like tree branches, those airways break off into four, then eight, then hundreds of thousands of little airways in each lung. Those are your bronchi.
At the end of each airway are tiny sacs called alveoli. Think of them as leaves at the end of the branches. Healthy lungs have hundreds of millions of alveoli. Each is covered with a thin layer of fluid that keeps the tiny sacs open, so oxygen is absorbed and carbon dioxide is excreted.
Your lungs also have millions of tiny hairs called cilia. Cilia act like little brooms -- constantly moving back and forth to clear out nasty stuff that makes its way into your lungs with every breath.
Cigarette smoke, by the way, kills cilia, destroying the very mechanism that protects your lungs from toxins.
The last part of healthy breathing is the musculature that supports your lungs -- the diaphragm. This large muscle pulls air down into your lungs. By using your diaphragm, you can develop techniques to help you breathe deeper.