Nonduality is the state or condition of not being separate and distinct while appearing to be so. It is the condition which allows us to say that there is no true separation between ourselves and anyone else or anything else in the world, for instance. When we say things like “We are all one,” or “God is in all things,” we are asserting that the condition we call ‘reality’ is a nonduality.
Nondualism is the systematic description of nonduality, or the systematic practice of nonduality.
More Complicated Answer:
Nonduality is the state that one arrives at when all distinctions and relations between ‘things’ are removed. Nonduality is thus a simple wholeness, rather than an “all in one” whole. It is very difficult to clearly contemplate such a simple wholeness, because by thinking about it and ‘conceptualizing’ it, we have lost the simple wholeness that is the real nonduality that we were trying to grasp.
Nondualism according to this understanding is an error.
The ‘As True As You Are Going to Get’ Answer:
Nonduality is ineffable.
Nondualism according to this understanding must be an apophatic* performance. In order to say that Nonduality is ineffable, we first posit 'Nonduality', making ‘Nonduality’ a creature of reason and thus positively identifying ‘Nonduality’ as some thing that can be reasoned about, and then in the same breath we take away this assertion by adding that this ‘what’ of which we speak is ineffable and thus beyond the reach of reason. The point is that Nonduality is not nothing, because then we would not even speak of ‘it’; but it is not something either, because if it were it could not be Nonduality; yet it is all things and no thing itself. Thus the name "Nonduality" is used to indicate a denial of multiplicity, yet the mind, seeing this denial may assume that it means 'one' as that is the opposite of multiplicity and in reasoning the mind is locked into certain forms of thought, amongst them the form of contradictories. But the name "Nonduality," while it denies multiplicity, also denies its contradiction and subsumes both. These words are an apophatic performance. If you can ‘see’ their meaning, you do not need any more definitions.
* Apophasis - the Greek designation for language that 'speaks away' or 'unsays' what it first affirms.