Ignorance is also called incorrect or incomplete knowledge. Avidyā or the synonym ajñāna is the individual (vyaṣṭi) aspect of māyā that manifests itself in the jāgrad avasthā (waking state) and the svapna avasthā (dream state) of the jīva, while it remains unmanifested and undifferentiated in the suṣupti avasthā (deep sleep).

Ātmā, the self, with the avidyā upādhi (the apparent, but not real, conditioning by ignorance) is known as the jiva, the individual. Brahman with the māyā upādhi, is īśvaraḥ. Of course, brahman and atmā are one and the same truth.

Ignorance is a specific state of knowledge of the mind in which tamas predominates. Thus āvaraṇa śakti, the concealing power “covers” knowledge. Jñāna-śakti with his quality sattva reveals knowledge of myself as free, full of consciousness. Consciousness, pure, unlimited, attribute-less knowledge is already there, but must be discovered through a means of knowledge such as vedānta. So that pure knowledge may shine (bhāti). Therefore, sat cit ānanda de taittirīya upaniṣad (Brahm ānandavalli 2.1) is listed as satyaṃ jñānam anantaṃ brahma, brahman is truth/existence, pure knowledge, boundlessness. Cit and jñānam are synonymous here. And ānanda (bliss) and ananta (boundlessness) are synonyms here.

It is not just an absence of knowledge. Ignorance is the opposite of knowledge, and is therefore a form of knowledge which is overcome only by right knowledge.

Ignorance hides that which is true and causes the projection of something else in its place. That “something else,” that erroneous or incomplete knowledge, when it is true, is opposed to that which is true.

That erroneous or false conclusion takes place in the mind, in vedānta, called the inner instrument (antaḥkaraṇa). Enlightenment is therefore only for the mind. This needs to be made even more precise. One of the four functions of the inner instrument is the intellect. There it is determined how someone thinks. Knowledge therefore takes place through or in the intellect. The intellect thinks, “It is true that I am the light of Brahman,” or the intellect thinks, “I am a miserable person.” In both cases, the belief (saṅkalpa) determines what my life looks like, as a reflection of divine light, or as a projection of low self-esteem and sadness. So that's how quickly your life can change when ignorance is replaced by its opposite knowledge. Basically, we are talking about an immediate shift.

Since the mind (manas) doubts (vikalpa), and thus will fiddle with self-knowledge, a well-developed intellect is needed for knowledge (vidyā, jñāna) and thus for freedom.

Ignorance cannot exist in atmā, because ātmā is impeccable, pure consciousness, knowledge. Ignorance occurs only in the jīva and is held in the mind. The jīva is an apparent union of mind and spirit. When it is recognized that the jīva status is no more than appearance, the ignorance of it also disappears. Thus, in ceasing to identify with the mind (and the body), the sages do not see ignorance, although they do recognize it in the apparitions around them.

In the end, there is no ignorance, no mind, and no māyā. If you come to a wrong conclusion because of avidyā, you come to adhyāsa, a wrong assessment of objects (including people). It is adhyāsa, misconception, not seeing things as they are, that is the direct cause of saṁsāra, while ignorance is of course the fundamental or ultimate cause (mūla avidyā). Ignorance is beginningless, but not endless. When a mind is clear enough to understand self-knowledge, hears the truth about itself, ignorance will disappear, the knowledge that removed ignorance will also disappear, and the self will shine, free as itself.

Adi Śaṅkara puts it this way in his bhāṣya (commentary) on the brahmasutras: svarūpāparijñānam ajñānam. Ignorance (ajñāna) is the not-knowing (aparijñāna) of one's true nature (svarūpa).

Another formal definition of ignorance: avyaktānirvacanīyā triguṇātmikā māyā yā prakaṭābhāsikā, sā eva ajñānam. Ajñānam/avidyā (ignorance) is māyā, which is unmanifested (avyakta), indefinable (anirvacanīya), composed of the three guṇas (sattva, rajas and tamas), and which appears as apparent reality (prakaṭābhāsikā). Prakaṭā from Sanskrit root kaṭ, appearing, manifesting.

Finally, this brief definition: tattvānyathāgrahaṇam ajñānam, ignorance (ajñāna) is the fallacy (anyathāgrahaṇa) of reality (tattva).