Nirvikalpa is a Sanskrit adjective with the general sense of "not admitting an alternative", formed by applying the contra-existential prepositional prefix निः ("away, without, not") to the term विकल्प ("alternative, variant thought or conception").
In Hinduism, when used as a technical term in Raja Yoga, the phrase nirvikalpa samādhi refers to a particular type of samādhi that has been practiced by many in India, and most recently experienced by Ramakrishna Paramahamsa. Recently Heinrich Zimmer in his book distinguishes Nirvikalpa Samadhi from other states as follows:
Nirvikalpa samādhi, on the other hand, absorption without self-consciousness, is a mergence of the mental activity (cittavṛtti) in the Self, to such a degree, or in such a way, that the distinction (vikalpa) of knower, act of knowing, and object known becomes dissolved — as waves vanish in water, and as foam vanishes into the sea. The difference to the other samadhis is that there is no return from this samadhi into lower states of consciousness. Therefore this is the only true final Enlightenment.
To put it simply. Nirvikalpa is a state in which the practitioner realizes that he/she is one with God. A state of non-duality.
This one word 'Nirvikalpa' describes the whole truth, that the God is within you. As he is everywhere and infinite, he is in you as well.
Nirvikalpa state may be achieved by samadhi meditation. But that the God is within oneself is described multiple times in Rig, Yajur, Sama Vedas and Puranas.
Nirvikalpa yoga state is achieved by vedic pundits who perform Mahanyasapurvaka Rudra Abhishekam. It is said "na Rudro rudram archayet" (meaning do not worship Rudra without you becoming Rudra).
In Lalitha Sahasranama Stotra, it is given that " Aham ityeva vibhavaye". Meaning, "Yourself is the Godess"
Shakti practitioners realize Godess Kali (Tara, Durga etc.) within themselves.
Nirvikalpaka yoga is a technical term in the philosophical system of Shaivism, in which there is a complete identification of the "I" and Shiva, in which the very concepts of name and form disappear and Shiva alone is experienced as the real Self. In that system, this experience occurs when there is complete cessation of all thought-constructs.
In Buddhist philosophy, the technical term nirvikalpa-jñāna is translated by Edward Conze as "undifferentiated cognition". Conze notes that only the actual experience of nirvikalpa-jñāna can prove the reports given of it in scriptures. He describes the term as used in Buddhist context as follows:
The "undiscriminate cognition" knows first the unreality of all objects, then realizes that without them also the knowledge itself falls to the ground, and finally directly intuits the supreme reality. Great efforts are made to maintain the paradoxical nature of this gnosis. Though without concepts, judgements and discrimination, it is nevertheless not just mere thoughtlessness. It is neither a cognition nor a non-cognition; its basis is neither thought nor non-thought.... There is here no duality of subject and object. The cognition is not different from that which is cognized, but completely identical with it.
A different sense in Buddhist usage occurs in the Sanskrit expression nirvikalpayati (Pali: nibbikappa) that means "makes free from uncertainty (or false discrimination) = distinguishes, considers carefully.
See also 
- Apte, p.555; Monier-Williams, p.542
- Usharbudh Arya translates it as "non-discursive" when applied to the subject of thought.Arya 1986, p. 111.
- For quotation regarding distinction as a type of samādhi, see: Zimmer 1951, pp. 436–437.
- For definition of Nirvikalpaka yoga as used in Kashmir Śaiva usage, see: Singh 1979, p. xxxiii.
- For nirvikalpa-jñāna as "undifferentiated cognition", see: Conze 1962, p. 253.
- For quotation including the translation "undiscriminate cognition" see: Conze 1962, p. 253, footnote ‡.
- For Buddhist usage as "makes free from uncertainty (or false discrimination) = distinguishes, considers carefully, and note that the term means "free from vikalpa", and Pali equivalent nibbikappa, see: Edgerton 1953, p. 304, volume 2.
- Arya, Usharbudh (1986), Yoga-Sūtras of Patañjali (Volume 1 ed.), Honesdale, Pennsylvania: The Himilayan International Institute, ISBN 0-89389-092-8
- Conze, Edward (1962), Buddhist Thought In India (First Ann Arbor Edition, The University of Michigan Press 1967 ed.), George Allen & Unwin Ltd., ISBN 0-472-06129-1
- Edgerton, Franklin (1953), Buddhist Hybrid Sanskrit Grammar and Dictionary (Reprint, Two-volume ed.), Delhi: Motilal Banarsidass, ISBN 81-208-0997-1
- Zimmer, Heinrich (1951), Philosophies of India (Ninth Bollingen Paperback, 1989 ed.), Princeton: Princeton University Press, ISBN 0-691-01758-1
- Comans, Michael. "The Question of the Importance of Samadhi in Modern and Classical Advaita Vedanta- Notes: 19", realization.org, 1993
- The Method of Early Advaita Vedanta p.141-4 on Nirvikalpa Samadhi