Nonduality in Buddhism
In the Chittamatra (mind-only) system of Indian Buddhist tenets, nondual refers to the absence (voidness) of the appearance (object) aspect of a cognition and the cognitive aspect of the cognition coming from different natal sources (Skt. dravya). The natal source of something is that from which something is born and which produces it, like an oven for a loaf of bread, or a potter's wheel for a clay pot. The natal source for both the appearance of an object and the consciousness and accompanying mental factors that perceive it is a karmic tendency (bija, seed) imputed on the alayavijnana (all-encompassing foundation consciousness, storehouse consciousness). It is not that the appearance of an object derives from a separate "external" object as its natal source. "External" means having a separate natal source from the mind that perceives it. Objects and the consciousness of them appear to come from different natal sources, but that duality is like an illusion. It does not correspond with the true situation.
The Chittamatra system, however, considers that objects of bare perception (seeing, hearing, etc), consciousness of them, alayavijnana, as well as voidness (nonduality) all have true unimputed existence, which means that their existence as this or that is established by defining characteristics on their own sides, independently of their being imputed on a basis (unlike a category existing non-truly, as something imputable on individual items sharing certain defining characteristic features).
Non-Gelug Madhyamaka Systems
In the Sakya, Nyingma, and Kagyu Madhyamaka systems, nondual has a similar meaning. It refers to the absence of the appearance (object) aspect of a cognition and the cognitive aspect of the cognition having separate existence from each other, and coming from difference sources (though they do not use the term "natal source"). They define alayavijnana quite differently from Chittamatra, and each of these three Tibetan systems has its own individual explanation in terms of the clear light (subtlest) level of mind. The important distinction between these three systems and Chittamatra is that nothing has true unimputed existence.
Several masters from these three systems differentiate self-voidness and other-voidness. There are many explanations of the two. Self-voidness being beyond words and concepts refers to voidness being beyond the conceptually knowable categories of existent, nonexistent, both, and neither. Other-voidness being beyond words and concepts refers to it being a level of mind that is subtler than the conceptual one.
Gelug Prasangika System
In the Gelug Prasangika-Madhyamaka system, nondual refers to the absence of true findable existence. True findable existence is "dual," in the sense that if it were to exist, it would be a second type of existence other than the absence of true findable existence. Although a mind having the instincts of grasping for true existence gives rise to an appearance of true findable existence, that appearance does not correspond to anything real, since there is no such thing as true findable existence. "True findable existence" means existence established by findable defining characteristics on the side of an object.
Both the Sakya, Nyingma, and Kagyu Madhyamaka and the Gelug Prasangika-Madhyamaka views derive from Nagarjuna. None of them say that everything is exactly the same as an illusion. Shantideva clearly points out that there is a distinction between killing a man and killing an illusion of a man. The non-Gelug systems, however, assert that everything that appears to a mind infected with unawareness (ignorance) is an illusion. These systems assert that because, unlike Gelug, they do not differentiate between the appearance of what something is and the appearance of how it exists. Because everything that appears to a mind mixed with unawareness appears to be truly existent, all those appearances are an illusion. Gelug, on the other hand, asserts that all such appearances are merely like an illusion. How they appear to exist is like an illusion in that it doesn’t correspond to how things actually exist, but their valid conventional appearance is not an illusion.
Advaita Vedanta shares some terminology with the Madhyamaka systems, and thus appears to have many similarities. However, Shankara's system has many crucial differences:
The Conventional Self
Shankara asserts that the conventionally existent, empirical living self (jiva) is the combination of a passive observer consciousness (sakshin) and an active inner physical organ (antahkarana) that assumes the form of objects cognized. Before liberation (moksha), the two are always together, although in deep sleep, the observer consciousness withdraws into ignorance (avidya), which is each living self’s share of illusion (maya). Buddhism asserts that the conventionally existent self is imputable on the aggregates, but is not the combination of any of them.
The Relation between the Self and Either Brahma or Voidness
Shankara asserts that each observer consciousness is identical with nirguna brahman (brahman without qualities), and that with liberation, the inner physical organ associated with that observer consciousness returns to maya-illusion, and that the observer consciousness merges with brahman without qualities. If we look in Buddhism at the relation of the conventionally existent self and self-voidness (the absence of true existence), Buddhism never says that the two are identical, nor that they merge with liberation or enlightenment. Self-voidness is the manner in which the conventional self exists, namely devoid of existing as a truly existent self. As for a truly existent self, there is no such thing. If we analyze in terms of other-voidness, and take other-voidness to refer to the clear light mind, then although, in general, both the appearance of a non-truly existent self and of a truly existent self are the appearance aspects of the clear-light mind, they are not identical with the clear light mind. "Identical" in Buddhism means totally the same. Although with enlightenment there is no longer an appearance of a truly existent self, let alone a belief in one, that does not mean that the truly existent self has merged with the clear light mind. Also, even with enlightenment, there is still a non-truly existent imputable self.
Shankara asserts that maya-illusion is the potency (shakti) in the God Ishvara. Empirical reality, perceived publicly by all, is the creation of Ishvara (ishvara-srshta) and is the practical commonsense world (vyavaharika). Private reality, perceived individually, is the creation of each living self (jiva-srshta) and is the apparent world (pratibhasika). Both the empirical world and the various private worlds spring from maya-illusion. I have no idea how Shankara's concept of srshta (springing, creation) compares with the Chittamatra discussion of natal sources and alayavijnana. I doubt that they are the same.
Maya-illusion in Advaita is not like the Samkhya assertion of existent primal matter (prakrti), with all objects being illusory (maya) perturbations (vikara) of primal matter.
According to Shankara, maya-illusion cannot be characterized as existent or nonexistent. Ignorance is when we confuse our private world with the public world. Objects perceived privately endure only so long as they are perceived by a particular living self, while objects perceived publicly endure even when no living self is perceiving them, since they are always perceived by Ishvara. Buddhism is very different from this, both in Chittamatra and Madhyamaka, as discussed above.
Space and Time
Shankara asserts that the various manifold living selves themselves, as well as maya-illusion itself, and Ishvara himself, are in neither space nor time. All publicly and privately perceived objects are in space and time, and space is in time. Time is what relates the various living selves and maya-illusion.
In Madhyamaka, time and space are not conceived as containers for objects. Time is a measurement of change, and space is the absence of anything tangible or obstructing that would prevent a material object from occupying three dimensions. Both time and space are devoid of true existence.
Brahman without qualities (nirguna brahman) displays or translates (vivarta, literally: turns) itself in the sphere of space and time as various objects springing from maya-illusion, as well as various individual living selves and Ishvara. However, brahman without qualities itself does not change. The unity (aikya) of all these diverse displays – in the sense of them all being displays springing from maya-illusion – is brahman with qualities (saguna brahma). Brahman without qualities, however, is nondual (advaita) in the sense of being beyond the differentiation of unity and diversity. Thus, Shankara's Advaita Vedanta theory is not the same as Ramanuja's Vishishta Vedanta theory of the evolution of maya-illusion and living selves from brahman as transformations (parinama) of it. Nor is it the same as the Kumarila's Purva Mimamsa theory of unity within diversity (bheda-abheda).
When Madhyamaka uses the argument of neither one nor many, this refers to the fact that if things were truly existent, such as the self and the aggregates, there would need to be either just one such thing (in which case the self and the aggregates would be the same thing, totally identical) or there would need to be several truly existent things, totally separate from each other. Since neither of these is logically the case, then there is no such thing as true existence. Voidness is the absence of true existence – the absence of there being either one or many truly existent things.