The alveoli are tiny sacs in the lungs where all the oxygen is absorbed, and all the carbon dioxide is released out of the bloodstream. A single air sac is called an alveolus, but you can't do much with just one microscopic air sac. So, medical folks almost always refer to them in the plural -- alveoli.
The walls of the alveoli are just one cell thick, and blood flows on the other side, allowing gasses to pass through the walls into the bloodstream while keeping blood out of the sacs. Conditions that cause blood to back up into the lungs, like congestive heart failure or a pulmonary embolism, can force blood across that thin wall into the alveoli themselves. The escaped blood will usually mix with air, and victims of this condition might cough up pink, frothy sputum.
Alveoli have the consistency of a rubber balloon, which allows them to stretch open to receive air from the outside and -- maybe more importantly -- collapse down and force air back out. Since the lungs are made of about 300 million of these little sacs, parts of the lungs can work even when some areas of the lungs have been damaged.
When alveoli have been damaged by disease or by toxins (like smoking), they will often lose their elasticity, and some will be destroyed completely. This is called emphysema, a type of COPD. When exhaling, the air remains trapped within enlarged air spaces due to the concurrent narrowing of the large airways in smokers.
Anything that interferes with the flow of air into and out of the alveoli, or interferes with the exchange of gasses across the alveolar walls, causes shortness of breath.
