Sankara's monism is a form of Hindu philosophy. In fact, it is one of the most popular ones. It emerges in the Hindu Scriptures and was developed by the Hindu philosopher and theologian Sankara in the 8th century. Sankara, it has been argued, made the theory cogent and intelligible, consistent with other scriptural claims and even used it as a tool in an attempt to try and convert recalcitrant Buddhists.
Briefly, Sankara's monism, or Advaita as it is called, says that there is no distinction between mind and body, and besides, that there is only one mind in the universe. This non-dual mind is identified with our Self, which is our thinking soul, and also with god. Everything else, including your and my individuality, is an illusion, called Maya, in this mind due to its intrinsic ignorance about the nature of things. There need be no cause for ignorance, Sankara argues, as it occurs naturally. Only wisdom needs to be explained. Thus there is no need to posit any further entities to back this ontological thesis.
Now, post-vedic Hindu scriptures contain ample references also to the doctrine of karmic rebirth.1 That is, every being is caught in an unending cycle of births and deaths. As you sow, so you shall reap. Thus, our "sins"2 will eventually catch up with us causing us (our souls) to be reborn in our next lives and eventually (after an infinite time) experience exactly what it is we did to other creatures in this and our previous lives. This has been proposed by some as the Hindu solution to the problem of evil. A very simplistic interpretation of this doctrine (folk-rendition) says that if you kill a butterfly in this life, then you will be reborn as a butterfly in one of your future lives during which this butterfly will be born as a human who will kill you, thus evening things out in the final reckoning.
At first sight, this doctrine itself seems fraught with many serious problems of personal identity and the final equalisation seems extremely doubtful given the interconnectedness of so many events. Much work has been done towards trying to make this theory more sophisticated, including the claim that an infinite regress does not merely postpone ethical compensation indefinitely, but achieves it. But none of them seems convincing to one who is not already convinced by its naive presentation above. I hope, however, that I am able to give a cogent rendition of the doctrine in this note that will solve all these problems convincingly.
Besides having problems of its own, the doctrine of karmic rebirth is also inconsistent with Advaita. Simply, if there is a soul that undergoes death and rebirth because of its actions towards other souls, then by implication there are many souls, each undergoing a series of regenerations. Each has its own bank of karmic consequences and each tries to work its way to salvation separately. Where then is Sankara's grand singular soul, the Atman, which is forever unchanging and blissful? If Sankara's Atman does not undergo birth or death as he claims3where then is the individual soul which undergoes karmic rebirth? As far as I can tell, Sankara sidesteps the problem as much as possible, and when the going gets really tough from the Buddhists, he circumvents it by claiming that there are two kinds of souls, the absolute, and the manifested. It is the manifested soul that is caught in the endless cycle of karmic rebirth - the absolute soul is free from karmic consequences and as such is untouched by birth or death. The apparent existence of the manifested soul is a consequence of Maya in the absolute soul.
I have never been truly happy with this interpretation. It looks to me to be suspiciously close to an ad hoc modification to Sankara's original thesis. I was also unhappy with this because there is a much nicer solution to this whole problem which goes as follows: We start off by assuming that there are indeed several individual souls as required by the doctrine of karma. There is no mention of any temporal parameters for the karmic law in the Hindu scriptures. Thus, the phenomenon of rebirth could be said to be temporally unbound. That is, while our bodies may be subject to temporal causes and effects, there is no reason whatsoever to suppose that our souls also are. Our souls may well be aspatio-temporal, i.e., beyond space and time. Thus, a person who dies today, may not necessarily be reborn tomorrow or in the next instant; he or she may be reborn a million years into the future or even (swallow this for the moment) in the past!
From this, it is but a small step, to propose that not only can the individual soul be reborn in the past and the future, it can even be reborn in the present! Thus, my neighbour with whom I am carrying on a conversation could be my own self reborn (or vice versa) and his or her neighbour and friend could all be rebirths of my soul. If we extend this scenario enough, a final picture will force itself on us, which also, by the way, provides ground for good ethics on pragmatic grounds. There are no other souls than our own. Every other soul is just a rebirth of this one soul at a different time and place. Notice also, how things even out very nicely in the final reckoning. There need even be no final reckoning. As long as we consider that every sentient being is just our own soul reborn at a different point in space-time, everything is always in a state of balance. What you do to your neighbour you are doing to yourself. Christ's ethical dictum "Do unto others what you would like others to do unto you" now becomes pragmatically founded. When you kill a butterfly, you have just killed yourself as a butterfly. You create and spend the karmic consequence simultaneously. You start off with a nil balance and you end with a nil balance in your karmic bank. Indeed, you are always in a state of no debits and no credits. The real individual soul, of which there is but one, can now be identified with Sankara's grand universal soul. The atman is the Atman. It is perpetually in the state of equanimity.
Notes
1 It is noteworthy that rebirth in the sense of either metempsychosis or transmigration was not part of vedic hindu religion. This is discussed in greater detail by R.E.Hume in his introductory essay to the thirteen principal Upanishads, 2nd ed. OUP, 1931.
2 strictly speaking all actions, but sins (pavam) in popular rendition.
3 cf. Mandukya Karika II.32. Sankara considers this verse to be a summary of the chapter on unreality.