The breathing in and breathing out is what is called here Mechanical Respiration. As human nature is, we take this as granted and do not give much thought to it but is one of the most wonderful systems not well understood by most of the people.
Lets see how does the breathing happen. From nostrils, the air gets filtered by hair and passes into upper nasal pharynx cavity to larynx mouth cavity to wind pipe to trachea to bronchial tree on each side to alveolar sacks. On its way it passes the vocal chords, thyroid glands and ventilates sinus cavity above nose. The respiratory system is part of thoracic cavity and separates from abdominal cavity by diaphragm, which is attached to lungs. and plays an important role in breathing.
So how does the Oxygen rich air flow in? The Boyle's Law states that PxV=RT, which means if we increase the volume of lungs, pressure will reduce inside lungs and air will be sucked by itself from outside.
Nature has provided following levers, in order from maximum leverage to minimal leverage, to do the breathing:
a. Diaphragm - when it moves down, or when stomach is expanded, air rushes in. Only babies use this and one can see their bellies going up and down when they are sleeping. But as they start to walk, this habit is lost. It is used very minimally when doing things strenuous.
b. Intercostal Muscles - lungs are attached to the rib cage via these muscles and by expanding chest, the air flows in.
c. Interclavicle Muscles - muscles attached to the collar bones and lungs which can be exercised when lifting one's shoulder, though very subtle movement of air happens, and one may not sense it if not made sensitized to air flows in the respiratory system.
d. Thoracic Muscles - by contracting and expanding, near vocal areas, one can also breath and is used in strenuous exercises.
e. Nasal Muscles - by contracting and expanding muscles in the nose, one can also breath, though people may not have much control unless these muscles are developed.
It is clear that we hardly exercise all muscles to maximize breathing or do anything consciously to use some of the muscles. To maximize metabolism, one can employ all knobs and to minimize metabolism, one can only use the nostril, at least theoretically! Once metabolism is in control, heart, mind, brain, etc., are all in control!
Please note in the thoracic cavity, there is a thymus gland, which is part of the lymphatic system, and connected via the largest lymphatic vessels called thoracic duct, complemented by right lymphatic duct. Breathing plays an important part in controlling the flow of lymphocytes from Thymus and plays a crucial role in Immune System. Thymus controls the production of T-Cells which play an important part of the immune system.
The lymphatic system has three interrelated functions: (1) removal of excess fluids from body tissues, (2) absorption of fatty acids and subsequent transport of fat, as chyle, to the circulatory system and, (3) production of immune cells such as lymphocytes (e.g. antibody producing plasma cells like T-Cells and B-Cells) and monocytes.
Breathing has 4 parts: (1) Inhalation - Purnika, (2) Inner Retention - Antriya Kumbhka, (3) Exhalation - Rachika, and (4) External Retention - Bahriya Kumbhka.
Permutation and combination of the four parts of the breathing components, along usage of different muscles in the respiratory system, and passage locks developed during the course of Yoga training, and Aasanas make an important part of Pranayama.
As discussed earlier, the systematic and conscious control of breathing can be used to control the mind and the metabolism of the body. It is rigorously can be used in Meditation and control of Mind.
The ultimate in Pranayama and Kumbhaka is called Kevala Kumbha, where one suspends the mechanical breathing in and out, with helps of mechanical, aasan and pressure locks, but only sustaining himself with natural ventilation and exchange of oxygen in nostrils. There are many degrees of it, depending on mental state, posture, place and time. This can be used to transcend Mind and is the goal of Raja Yoga.
Over the years of Yoga practice, one develops control of locks to aid in this process. Please be advised, do not try this without guidance as death and damage can result. A lot of magic powers have been mentioned by texts on this aspect, but I do not know about them, except of some calming, healing and meditative powers. Even if one has mastered this, please make sure that you allow your body to recover after doing this Kumbha, before doing any activity, like walking or getting up, as you may faint or pass out.
In Geeta in Chapter 6, On Meditation, and other chapters too, it is mentioned that one should do Metitation by suspending the Apnaa and Pranaa and the interpretations given by most of the famous translators is true and subject to their personal experiences - and in one journey one may experience all of them. References: (1) Kevala Kumbha and (2) Dr. Sivananda on Kevala Kumbha. Apnaa is the downward movement of the Life Forces and Pranaa is the upward movement of the Life Forces - and the two are having a grosser representation in the Inhalation and Exhalation Process.
Geeta says:
Apane juhvati pranam pranepanam tathapare; Pranapanagatee ruddhva pranayamaparayanah (Gita, Ch. IV-29.). Others offer Prana (outgoing breath) in Apana (incoming breath) and Apana in Prana, restraining the passage of Prana and Apana, absorbed in Pranayama. Pranayama is a precious Yajna (sacrifice). Some practise the kind of Pranayama called Puraka (filling in). Some practise the kind of Pranayama called Rechaka (emptying). Some are engaged in the practice of Pranayama called Kumbhaka, by impeding the outward passage of air, through the nostrils and the mouth, and by impeding the inward passage of the air, in the opposite direction. Reference: Geeta on Kevala Kumbha - Translation by Dr. Sivananda.