What is Advaita?
"You philosophers are a bunch of fools" said the theologian. "You are like blind men in a dark room looking for a black cat that isn't there!" "That may be true" replied the philosopher, "but you theologians would have found it."
Advaita is a school of Hindu philosophy. The greatest proponent of Advaita in Hindu history can be said to be the South Indian philosopher Sankara, who lived in the 8th century. Interestingly, Sankara had employed much the same method as Descartes did in the 18th century to argue that the mind is different from its body. However, the implications he drew from his argument were quite different to those of Descartes.
Descartes used the Leibniz identity princple to argue that Mind and Body (brain) are distinct entities, since at least one of them possesses a property that the other doesn't. That is, the body possesses the property that its existence can be doubted, but the Mind doesn't - Cogito ergo sum (I think, therefore I am). The very act of doubting the existence of a Mind is a mental act which necessarily implies the Mind's existence. Since the existence of the Mind is self-evident, but that of the body isn't, they must be distinct entities. Hence Cartesian dualism.
Sankara, on the other hand, had used this same argument, but went on to argue that since only the Mind's existence can be established without doubt, and not that of the body, one should rationally only consider the Mind as real. Properly, the body should be dismissed away as unsubstantiated (a product of illusion or Maya). In Sankara's terminology, the Mind is called the Atman which roughly translates to Self or Soul.