In the teachings of the Twentieth century Bengali philosopher-sage Sri Aurobindo, Integral yoga (or purna yoga, Sanskrit for full or complete yoga, sometimes also called supramental yoga) refers to the process of the union of all the parts of one's being with the Divine, and the transmutation of all of their jarring elements into a harmonious state of higher divine consciousness and existence.
Integral Yoga must not be confused with a trademark "Integral Yoga" of Swami Satchidananda.
Sri Aurobindo initiated and defined integral yoga in the early 1900s as "a path of integral seeking of the Divine by which all that we are is in the end liberated out of the Ignorance and its undivine formations into a truth beyond the Mind, a truth not only of highest spiritual status but of a dynamic spiritual self-manifestation in the universe." 
He describes the nature and practice of integral yoga in his opus The Synthesis of Yoga. As the title of that work indicates, his integral yoga is a yoga of synthesis, intended to harmonize the paths of karma, jnana, and bhakti yoga as described in the Bhagavad Gita. It can also be considered a synthesis between Vedanta and Tantra, and even between Eastern and Western approaches to spirituality.
Textual sources: The theory and practice of Integral Yoga is described in several works by Sri Aurobindo. His book The Synthesis of Yoga, the first version of which appeared in the Arya, was written as a practical guide, and covers all aspects of Integral Yoga. Additional and revised material is found in several of the later chapters of The Life Divine and in other works. Later, his replies to letters and queries by disciples (mostly written during the early 1930s) were collected into a series of volumes, the Letters on Yoga. There is also Sri Aurobindo's personal diary of his yogic experiences, written during the period from 1909 to 1927, and only published under the title Record of Yoga.
No definitive method
Whereas Sri Aurobindo and the Mother taught that surrendering to the ‘higher' consciousness was one of the most important processes of the supramental yoga, neither established a universal definitive method for every practitioner of the yoga, due to the individual differences. Both left the open-ended question as to how the supramental consciousness would act and establish itself in Earthly life.
The aim of integral yoga: Integral development
Most yogas, except such paths as Natya Yoga, only develop a single aspect of the being, and have as their aim a state of liberation or transcendence. But the aim of integral yoga is the transformation of the entire being. Because of this, the various elements of one's make-up - Physical, Vital, Mental, Psychic, and Spiritual, and the means of their transformation, are described in great detail by Sri Aurobindo, who in this way formulates an entire integral psychology. The goal is then the transformation of the entire nature of one's being. Nothing is left behind.
The process...accepts our nature...and compels all to undergo a divine change...In that ever progressive experience, we begin to perceive how this lower manifestation is constituted and that everything in it, however seemingly deformed or petty or vile, is the imperfect figure of some element in the divine nature. – Sri Aurobindo, Synthesis of Yoga, p.47)
Also distinguishing Sri Aurobindo's teaching from most other mystical paths is the need for transformation of the personal and relative nature. So the integral yoga is twofold; both a spiritual realisation of God or Transcendence or Enlightenment, and, through this, a complete change and transformation of both the inner and the outer nature. Through this double action, one is thus made able and fit to manifest a divine consciousness, and in this way becomes part of a divine work.
The Realisation of Supermind
Sri Aurobindo considered man's present mental consciousness to be a transitional stage in terrestrial evolution, and that our civilization is at the brink of an evolutionary leap or shift towards a greater or ‘supramental' experience and capacity.
With regard to supermind and mind Sri Aurobindo wrote, ‘There is an eternal dynamic Truth-consciousness beyond mind; this is what we call supermind or gnosis. For mind is or can be a truth seeker, but not truth-conscious in its inherent nature; its original stuff is made not of knowledge, but of ignorance.' 
Sri Aurobindo considered the supermind to be an all-organizing and all-coordinating principle of truth-consciousness secretly involved in the material creation and he saw its emergence as the next logical and inevitable step in terrestrial evolution.