In the Vedantic (and subsequently Yogic) schools of Hinduism, Brahman is the name given to the concept of the unchanging, infinite, immanent and transcendent reality that is the Divine Ground of all being. It is regarded as the source and sum of the cosmos, that constricted by time, space, and causation, as pure being, to some extent a sort of world soul. In the beginning of its history, it was deemed a sort of super-stuff from which all that is arises, and debuts with this verse:
Great indeed are the Gods who have sprung out of Brahman. - Atharva Veda
However as the centuries passed and Upanishads, the primary Vedantic scriptures, are written the concept of Brahman fittingly grew in scope and complexity. Soon, the ancient writers of the Upanishads, around the 1st millennium BCE, insisted that Brahman, in addition to being material, efficient, formal and final causes of the cosmos, was also beyond all four senses of origin. Essentially, it is also beyond being and non-being alike, and thus does not quite fit with the usual connotations of the word God and even the concept of monism. It is said that Brahman cannot be known, that we cannot be made conscious of it, because Brahman is our very consciousness. Brahman is also not restricted to the usual dimensional perspectives of being, and thus enlightenment, Moksha, Yoga, Samadhi, Nirvana, etc. in the Hindu perspective is not merely coming to know Brahman, but to realize one's 'Brahman-hood', to actually realize that one is and always was Brahman. Indeed, closely related to the Self concept of Brahman is the idea that it is synonymous with jiva-atma, or individual souls, our atman (or soul) being readily identifiable with the greater soul of Brahman.
Etymology and origin of the name Brahman
In Vedic Sanskrit, Brahman (neuter gender) means "growth", "development", "swelling", also "pious utterance", "worship", from a root bṛh, "swell", cognate to English "bulge". brahmán (masculine gender) on the other hand, nowadays replaced by brahmin, is a priest, one of the Brahmin caste (brahmin formerly being the adjective relating to brahman).
Connected with the ritual of pre-Vedantic Hinduism, bráhman signified the power to grow, the expansive and self-altering process of ritual and sacrifice, often visually realized in the sputtering of flames as they received the all important ghee (clarified butter) and rose in concert with the mantras of the Vedas. Brahmin came to refer to the highest of four castes, the Brahmins, who by virtue of their purity and priesthood are held to have such powers.
An minority opinion in historical linguistics and the Sanskrit community is that of Georges Dumézil, who posits that etymologically, the Latin word flamen (a priest) may be cognate to brahman, an idea that would support his "trifunctional hypothesis".
The traditional mystic phrase that is seen to be the only possible description of Brahman as we, with limited minds and being, can entertain is "Sacchidananda". Broken down, it is "sat", "chitta", and "ananda" meaning "truth", "consciousness", and "bliss" respectively.
Brahman and Atman
Philosopher mystics of the Upanishads identify Brahman with atman, the inner essence of the human being, or the human soul. In the Hindu pantheon, Brahman should not be confused with the first of the Hindu trinity of Brahma (the Creator), Vishnu (the Preserver) and Shiva (the Destroyer). Brahma is, like the other gods, Ishwar, or manifested Brahman, fundamentally ego-conscious, whereas Brahman is without ego, without existence and beyond form.
Enlightenment and Brahman
While Brahman lies behind the sum total of the objective universe, some human minds boggle at any attempt to explain it with only the tools provided by reason. Brahman is beyond the senses, beyond the mind, beyond intelligence, beyond imagination. Indeed, the highest idea is that Brahman is beyond both existence and non-existence, transcending and including time, causation and space, and thus cannot ever be known in the same sense as one traditionally 'understands' a given concept or object.
Hindus also regard Brahman as the all pervading consciousness which is believed to be the basis of all the animate and inanimate entities and material. It is believed that the universe is not just conscious, but that universe is consciousness, and this consciousness is Brahman. Hindus believe that human consciousness has forgotten its identity, that of Brahman, as if a drop of water from a vast ocean thought itself separate, and that the only path to merge back into that Brahman or supreme consciousness is through the paths of devotion, moral living, and/or meditation, often expressed in various systems of Hindu spiritual practices known as Yogas.
If one seeks Brahman, Atman seeks truth and accepts it no matter what it is. Atman accepts all truths of the self/ego, and thus is able to accept the fact that it is not separate from its surroundings. Then Atman is permanently absorbed into Brahman. This is how one forever escapes rebirth.