It would be a good to first clarify some of the terms one often hears often but gets confused.
Yoga means to yoke with the Divine - the First Principle of your existence! Actually Yoke is nothing but goes back to the word Yoga. "K" and "g" are Guttural sounds and "a" and "e" can be confused in transliteration of sounds from one language to another language. Yoga basically aims to having zero mental modification - calming of the mind and transcending of the mind - and touching the core of your existence - the Soul or the First Principle.
Pranaa-yama means control of life current, since the word "yama" means control and the word "pranaa" means "life current". This life current is what makes up the sum total Physiological, Psychological forces, etc., of which we know some of them, like breath and respiration, metabolism, mind, etc.
It is one of the important steps of Yoga Philosophy of India, in realization of one's Self! In a way it is complementary to the Philosophy of Vedanta. The prior is a practical approach and the later is a theoretical approach.
Vedanta is a theory on meta-physics on Consciousness and what constitutes a Self and helps one stretch his or her identification with microcosm of Body, Senses, Mind, Ego and Intellect to the macrocosm of Universal Presence and Manifestation of the Consciousness called Brahman, which means which is ever growing present and stronger First Principle on which all other Scientific and Material Laws hold true. It is that summum-bonum of all existences, animate or in-animate. This philosophy helps individual see the Unity in the Plurality and the Eternity of the Self.
Yoga is a very ancient Philosophy and has been found its presence as far in time as around 3000 BC, and as far flung as in Aum In Crete And America and Aum Sign In America! Please see my postings on Aum in Crete and Aum in America. In recorded History it was first outlined and documented by Pannani, which is called as Asta Anga Yoga.
Asta Anga Yoga has following eight steps: (1) Yama, (2) Niyama. (3) Aasans, (4) Pranayama, (5) Pratihaar. (6) Dharna, (7) Dhyana and (8) Samadhi. Yama is ethical and moral conduct and Niyama is practice and observations. Pranayama we know. Dharna is basically a "single minded flow of thought on an object, thought, sound, smell, visualization and tactical senses". For now we will not go into details of these words. Dhyana is basically loosing of the "I" or "Ego" principle which one experiences in a deep sleep state but here one is awake! Samadhi is basically nothing but a conscious slow and volunteered death in a belief for the total liberation and the merging with the Brahman to get one self out of the cycle of Life of Births and Deaths - as per the Hindu Philosophy of Reincarnation of Soul according to the Karma Theory.
Kundalani Yoga is nothing but an extreme form of Pranayama where one experiences a Physiological and Psychological Metamorphosis recorded by Western Medical Science also. It is said to be a double edge sword, and if not properly followed under the guidance of a learned and experienced Guru, one can loose one's self to a permanent insanity!
Hatha Yoga is nothing but a physical preparatory steps focusing on Asans and some Pranayama mainly, lesser than Kundalini Yoga, to reach Raja Yoga, which is Total Mind Control. So it is a subset of Astanga Yoga with more practical aspects.
The Mind is compared to a Charioteer, and the two Horses are Breath and Metabolism, which form the 3 legs of a stable stool called Body. Usually Mind becomes slave to thoughts and desires, conscious or unconscious, and hijacks the metabolism and the breath. So when one is angry for some reasons, his mind is very agitated, blood pressure, pulses and heart beats are high, and breath is very heavy and forceful. But when one is lost in the serenity of something creative, like painting, music, singing, writing, and even some focused, like the thinking of great thinkers, the breath almost stops and metabolism comes to its lowest operating point. Need less to say the prior is associated with stress and co-related to high levels of Cortisol (a form of Harmone produced in Adrenal Gland on top of kidneys) and stress related diseases touching all aspects of body. The biggest nerve complex is Vagus nerve arising from brain which touches most of internal organs, and diseases of these organs if the stress persists for long time without any abatement. Then the brain's endocrine system is also affected in a complex chain and feedback cycle, The cardiovascular, the lymphatic, the para sympathetic and the sympathetic systems are also affected. Body needs endorphins for its well being and well keep, besides other neuro-transmitters like Serotonin, which controls the moods, energy level, thoughts, sleep cycle, and the physiological process like hunger and sexual drive.
So the theory of Pranayama says that through the breath control, one can control the mind and metabolism! There is nothing new to people who have read Dale Carnegie's books on Art of Public Speaking and Don't Worry, Start Living, and Lamas classes for prenatal care. The Art of Leaving focuses on this aspect. At Harvard Medical School, and in one Nova show, they showed Tibetans Monks doing meditation, and controlling their brain waves and their body physiology. In fact they were are in snow, almost naked, and their bodies were covered with wet clothes, and then they showed steam coming out of their bodies!
Tantra Yoga has many parts, including touching parts of Kundalani Yoga, Vedanta in form of Kashmir Shaivism, and to the grosser form of Sexual Union Principle, found popularized in the West without much of understanding, and which is the basis of Khujaroah Temples, besides the Magic, Toxication and the Non Veg Part. More on this some other time!