What may seem obvious about breathing is actually overlooked and taken for granted by most Westerners. Breathing is absolutely essential to life itself and all others functions of the body depend on it. Without proper breathing technique, the body is not oxygenated effectively, which leads to many physiological and emotional problems. This "science of breathing" has been taught for thousands of years and is the foundation of yoga offering not only physical and mental benefits, but spiritual ones as well. Prananyama, which literally means "to extend the vital life force" is the formal practice of controlling the breath and the central practice in yoga for living a life of health, wellbeing, energy, and peace. Breath is one of the 4 main sources of prana, the others being food/water, meditation and sleep.
Life is breath and breath is life. Neither exists without the other. However, the way you breathe also determines the way you live, which makes this topic vitally important. By listening to your own breath or the breath of another, you can very quickly assess someone's mental and physical state of being. Most people who are unconscious of the breath take short breaths at the location of the upper chest or collarbone, also referred to as shallow breathing. This form of breathing requires the greatest expenditure of energy, robs the body of nutrients and manifests disease. It also has a strong effect on the energy within the body and depletes your strength and vitality. It is not surprising that so many people breathe this way. Civilization has employed improper methods, attitudes and habits on us in relation to how we walk, stand and sit, which greatly impacts our breathing. It is quite normal to retract our breathing or hold the breath when confronted with stressful situations or a continued state of anxiety. The fact that tension cannot exist alongside a relaxed form of breathing is motivation enough to take the time to learn and practice full, deep breath breathing.
Natural and correct breathing (referred to as diaphragmatic, abdominal or belly breathing) requires breathing through the nose, deeply taking air into the belly, abdomen and lungs. A proper breath is equal in length for both the inhale and exhale (or a bit longer on the exhale), with a moment of holding the breath for just a few seconds in between. With proper practice, breathing correctly can become unconscious, requiring little or no thought, but providing a lifetime of health benefits.
To appreciate the health aspects a bit more, it's important to understand that enough air needs to reach the lungs in order to purify the blood and provide the proper nourishment. With shallow breathing, this does not happen. Instead toxins continue to circulate in the bloodstream and throughout the body, which ultimately manifests into disease. Proper breathing activates the parasympathetic nervous system, which reduces stress and its effects on the body and mind. The mind becomes more focused and still, lessoning mental noise, and physical, emotional and spiritual blockages begin to clear. Breath is intimately connected to your ability to heal and breathing techniques can help one respond more quickly to any type of treatment. Oxygenated blood strengthens and vitalizes every tissue, muscle and organ in the body, replacing "tired" cells with new ones. Breathing is therefore not just about the movement of air, but also the movement of energy and critical to healing and preventing both mental and physical illness.
There are many resources online that explain breathing techniques that can assist you in altering your habits as well as address specific health concerns. Yoga also teaches prananyama where one can learn breathing exercises such as deergha swasam, rapid diaphragmatic breathing or nadi suddhi, alternative nostril breathing. If you are someone who would like to go a bit deeper, there are some very powerful courses focused primarily on breathing such as those taught worldwide by the Art of Living, a nonprofit organization. Whichever route you choose you are sure to notice the effects almost immediately as your body, mind and spirit begin to harmonize.
 Yogi Ramacharaka (1903). Science of Breath. Retrieved from http://www.hermetics.org/pdf/ScienceOfBreath.pdf
 Holcombe, K. (2012, August). Healing breath. Retrieved from http://www.yogajournal.com/practice/2844
 Mercola, J. (2000, February 20). Breathing exercises and self healing. Retrieved from http://articles.mercola.com/sites/articles/archive/2000/02/20/breathing-part-one.aspx