Sometimes VO2 max is referred to as maximal oxygen consumption, maximal oxygen uptake, peak oxygen uptake or aerobic capacity. Never mind all that fancy and technical lingo. It basically stands for the maximum amount of oxygen your body can take in and actually utilize at max exertion.
Made even simpler, if you increase your O2 uptake, you'll run faster, bike faster, and/or swim faster. Your performance in any aerobic endurance activity would improve with an increased or high VO2Max.
Simply breathing faster or heavier won't increase your VO2Max. We've all seen runners huffing and puffing who look like they're about to hyperventilate (and if we're being honest, we can all probably include ourselves in that group at one point or another).
Breathing for New Runners
New runners often experience rapid heavy breathing. As a new runner's heart rate increases, his/her body automatically knows it needs more oxygen, so the most natural thing to do is to breathe faster. The reason this isn't helpful is because the breaths are usually shallow and even though the runner is breathing rapidly, he's not getting the oxygen deep into the lungs where it's taken by the blood stream to the muscle to make energy. A new runner needs to concentrate on a deep inhale and a big exhale to clear the lungs of CO2. Focusing on belly breathing (seeing the abdomen expand with each inhale rather than then chest) will help new runners get more oxygen deep into the lungs.
The amount of air inhaled and exhaled with each breath is your tidal volume. It increases from .4 to 1L at rest to as much as 3L during aerobic exercise. An aerobically unfit person's tidal volume is going to be less than that of an aerobically fit person. Also, a person who's aerobically unfit isn't going to have a very complex network of capillaries (the smallest blood vessels woven throughout the muscle fibers) at the muscle level. A person who has a higher tidal volume and a more complex system of capillaries at the muscle level is going to be able to get more oxygen into the lungs which will diffuse into the pulmonary blood and make its way down to the capillary level to the many mitochondria in the muscle. Mitochondria are found inside the cells, and they're the power producers. The mitochondria is where cellular respiration occurs, which produces fuel for the cell's activities.
The good news is that you can increase your tidal volume, expand your capillary network, and increase the number of mitochondria you have. By doing consistent aerobic exercise (running, cycling, swimming, etc.), you'll increase all of the above. The more you run, your body will actually increase and build that important network of capillaries in the muscles and when that happens, you'll actually increase the number and size of the mitochondria in the tissue. When that happens your body will be more efficient at extracting oxygen from the blood and getting it to the muscle where it's needed for energy production. For experienced runners, more intense workouts such as intervals, hill repeats, fartleks, and/or tempo runs will help up their VO2Max.