Breath in Sanskrit is Prana. Prana also means life force, the fundamental energy of our being. Physical breath and energizing life force are inextricably intertwined.
Breath and life force are implicit in the words "hatha yoga." Ha (sun) and tha (moon) refer to the sun and moon nadis (energy channels) running along the spine, the central axis of the body. Prana as breath (inhalation, exhalation, and retention) regulates the flow of prana as energy throughout the body. Effective breathing during asanas leads to unobstructed, properly directed energy flow, and maintenance of healthy life force, throughout the entire body and mind.
Pranayama is breath management to facilitate physical, mental, and spiritual progress. It is one of the eight steps of Raja yoga (spiritual yoga, explained further on Patanjali page) and also a critical element of Hatha yoga.
The term is also used to mean breath utilization and control during performance of asanas, individually and in vinyasas. Yoga schools and teachers vary considerably in their approaches to breathing during asanas, from casual treatment to rigorous guidelines. Yoga practitioners and students must experiment and determine what works for them.
On a practical level, breathing deeply & regularly provide oxygenation of the body’s cells (muscles and other), reduction of cardiopulmonary strain (always a good idea), and improvement of brain function (good for everybody). Energy flow is improved, whether in western terms like electric current or eastern terms such as prana and qi.
Yoga breathing is through the nose. Guidelines for breathing during asanas generally follow two rules under the fundamental guidance of synchronizing breath with body motion. The first rule is inhalation (puraka) when the asana encourages expansion or outward thrusting of the chest, and exhalation (recaka) when the chest is compressed or drawn in. This seems to be common sense, although it is more difficult in practice than it sounds. The second rule is exhalation during exertion and inhalation after or between bursts of exertion. On occasion this is counter-intuitive, but in fact it can be quite effective.
Sometimes, for an individual, the two principles are in conflict for a particular asana. In this situation, comfort should be the deciding factor; this will usually involve following the principle of exhalation during exertion. When a position is being held for a length of time, the body can’t inhale or exhale indefinitely. Normal respiration should resume, utilizing the inhalation and exhalation to begin mini-rhythms of chest expansion/contraction and exertion/recuperation within the overall position being maintained.
Kumbhaka is retention of breath. Kumbhaka is a component of pranayama exercises. It also can be utilized for specific purposes in certain asanas.