Brahman
Brahman, in the Upanishad (Indian sacred writings), is the supreme existence or absolute reality. The etymology of the word, which is derived from Sanskrit, is uncertain. Though a variety of views are expressed in the Upanishad, it is concurred in the definition of Brahman as eternal, conscious, irreducible, infinite, omnipresent, and the spiritual core of the universe of finiteness and change.
The Brahman is not to be understood as a "Creator God" as in western concepts, but rather as a cosmic force of nature, much like the Dao in Daoism, the Monad of Gnosticism, The Great Spirit of Native American spirituality, or the Netjer of Egyptian spirituality.
The Upanishad contain four sentences, the Mahāvākya (Great Sayings), which were used by Shankara to establish the identity of Atman and Brahman as scriptural truth:
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"Prajñānam brahma" - "Consciousness is Brahman" (Aitareya Upanishad)
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"Aham brahmāsmi" - "I am Brahman" (Brihadaranyaka Upanishad)
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"Tat tvam asi" - "That Thou art" (Chandogya Upanishad)
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"Ayamātmā brahma" - "This Atman is Brahman" (Mandukya Upanishad)
Brahman should not to be confused with Brahma (one of the Hindu Trinity, the Trimurti). The Hindu pantheon of gods (deva) are mundane manifestations of Brahman. They are the personified attributes and qualities of Brahman, and should not be taken as gods to be worshipped as with other religions. The Deva, although blessed with longevity, still have a finite lifespan, and are subjected to rebirth. Even Brahma is not eternal, as he gets absorbed back into Purusha at the end of an aeon (kalpa), and is born again at the beginning of a new kalpa.
Paramatman
Paramatman means Supreme Soul. If the self of the body is the soul, then Paramatma is the self of the soul. That is, the soul of the soul is the supreme soul. It is the "Primordial Self" or the "Self Beyond" who is spiritually identical with the absolute and ultimate reality. Selflessness is the attribute of Paramatman, where all personality/individuality vanishes.
Atman
Atman, one of the most basic concepts in Hinduism. It is the essence of Brahman, but often is referred to as the "Self" or "soul", identical with the eternal core of the personality that after death either transmigrates to a new life, until oneself attains liberation (Moksha, also known as Nirvana) from the bonds of existence in the cycle of life & death (Samsara).
Jivatman
While the Atman is quality-less and attribute-less, when gets combined with Prakriti, forms different kinds of senses (both subtle and gross) and gets different material natures. This combination of Atman and qualities arising from Prakriti is called as Jivatman. It is Maya (Illusion) that clouds the Atman from Paramatman.
Purusha & Prakriti
Purusha is a complex concept whose meaning evolved in Vedic and Upanishadic times. Depending on source and historical timeline, it means the Cosmic Being or Self, Cosmic Awareness, and/pr Universal Principle.
In the Upanishad, the Purusha concept refers to the abstract essence of the Self, Spirit and the Universal Principle that is eternal, indestructible, without form, and all-pervasive.
In Samkhya philosophy, Purusha is the Cosmic Principle, Pure Consciousness, unattached and unrelated to anything, which is "nonactive, unchanging, eternal, and pure". It is the Purusha (Masculine Cosmic Consciousness) uniting with Prakṛti (Feminine Primordial Creative Energy - Nature), giving rise to life.
Kaivalya
Kaivalya, meaning "solitude" or "detachment", is the ultimate goal of Yoga. It is the liberation of Purusha (consciousness) from Prakriti (matter). The concept of liberation is one of the distinguishing features of Hinduism, Buddhism, Jainism and Sikhism. It is variously referred to as Mukti, Moksha, Kaivalya and Nirvana in the scriptures.
Kaivalya-Mukti is the detachment and independence from defilements leading to the cycle of birth and death. Yogi who reach this state is called a Kevalin.
The Kaivalya-Mukti is described in some Upanishad, such as Muktika and Kaivalya as the most superior form of Moksha which can grant liberation both within this life, as Jivanmukti, and after death, as Videhamukti.
There are 2 stages of Moksha:
The First Stage Of Moksha:
The first stage is when you experience freedom from unhappiness in this very life. If someone hurts us or if there is any mishap at home, even in such a situation if our happiness prevails, it is said that the person has practically experienced the first stage of Moksha.
Even in the midst of suffering or an adversity, you experience the bliss of the Self from within. This is possible when one attains Self-realization and remains in Pure Soul. When you become free from self-ignorance, then you achieve this stage, which is free from all unhappiness.
After Self-realization, no suffering will touch you. You will be able to see your own mistakes in the presence of the light of the awakened Soul.
The Second Stage Of Moksha:
The second stage of Moksha is attained when you become free from all your Karma, i.e. freedom from all attachments of worldly atoms. Not a single atom remains attached to your Soul. All the Karma are completely exhausted (cleaned) and you become free from the cycle of birth and death. In order to attain this stage, you need to have total and complete experience of the Soul and be in a human body. It is only with this body that you can see the whole Universe, including each and every atom of the Universe and then achieve ultimate liberation.
Ishvara
The root of the word Ishvara comes from "ish" meaning "capable of" and "owner, ruler, chief of". The second part of the word Ishvara is "vara", which means depending on context, "best, excellent, beautiful", "choice, wish, blessing, boon, gift", and "suitor, lover, one who solicits a girl in marriage". The composite word, Ishvara literally means "owner of best, beautiful", "ruler of choices, blessings, boons".
The word Ishvara does not appear in Rigveda. However, the verb "ish" does appear in Rigveda, where the context suggests that the meaning of it is "capable of, able to". It is absent in Samaveda, is rare in Atharvaveda, but it appears in Samhita of Yajurveda. The contextual meaning, however, is neither god nor supreme being.
The word Ishvara appears in numerous ancient Dharmasutra. However, there Ishvara does not mean God, but means Veda. Ishvara in Dharmasutra could alternatively mean "king", with the context literally asserting that the Dharmasutra are as important as Ishvara (the king) on matters of public importance".
Thus, across all the Vedic scriptures, Ishvara can be correctly understood as the embodiment of Brahman or the Veda.