Atman (or paramätman, the highest Self), for Advaita Vedanta is that pure, undifferentiated self-shinning consciousness, timeless, spaceless, and unthinkable, that is not different from Brahman and that underlies and supports the individual human person.

Ätman is pure, undifferentiated, self-shinning consciousness: It is a supreme power of awareness, transcendent to ordinary sensemental consciousness, aware only of the Oneness of being.

Atman is that state of conscious human being wherein the division of subject and object, which characterize ordinary consciousness, are overcome.   Nothing can condition this transcendental state of consciousness:   among those who have realized it, no doubts about it can arise.   Atman is thus void of differentiation,  but for Advaita it is not simply a void:  it is the infinite richness of spiritual being.

The full title of the philosophy is 'Advaita Vedanta'.   'Vedanta' simply means that it derives from the scriptures that form the last part of the Vedas, the four sacred texts of the Hindu religion.  The literal meaning is 'the end of knowledge', in the sense of being the highest knowledge one can attain.  It is not itself a religion, however—there are no churches or priests.  The first part of the Vedas does contain rituals and so on but Advaita does not itself rely on these.

Advaita is an extremely simple philosophy. Its complete essence is summed up in its Sanskrit name: a - not, dvaita - two.  In a very real sense, there is no need for a book to explain it.  It can be summed up in a single sentence.

            There are not two things.

It is necessary to understand that I Am,
In order that I may know that I Am Not,
So that, at last, I may realize that
I Am Not, therefore I Am

ANYTHING VISIBLE, and anything that can be grasped by thought, is bounded. Anything bounded is finite. Anything finite is undifferentiated. Conversely, the boundless is called Ein Sof, Infinite. Since it is boundless, there is nothing outside of it. Since it transcends and conceals itself, it is the essence of everything hidden and revealed. Since it is concealed, it is the root of faith and the root of rebellion. As it is written, “One who is righteous lives by his faith.” The philosophers acknowledge that we comprehend it only by way of no.

Emanating from Ein Sof are the ten sefirot. They constitute the process by which all things come into being and pass away. They energize every existent thing that can be quantified. Since all things come into being by means of the sefirot, they differ from one another, yet they all derive from one root. Everything is from Ein Sof, there is nothing outside of it.

One should avoid fashioning metaphors regarding Ein Sof, but in order to help you understand, you can compare Ein Sof to a candle from which hundreds of millions of other candles are kindled. Though some shine brighter than others, compared to the first light they are all the same, all deriving from that one source. The first light and all the others are, in effect, incomparable. Nor can their priority compare with its, for it surpasses them; their energy emanates from it. No change takes place in it - the energy of emanation simply manifests through differentiation.

Ein Sof cannot be conceived, certainly not expressed, though it is intimated in every thing, for there is nothing outside of it. No letter, no name, no writing, no thing can confine it. The witness testifying in writing that there is nothing outside of it is “I am that I am.”  Ein Sof has no will, no intention, no desire, no thought, no speech, no action – yet there is nothing outside of it.

A real haiku's gotta be as simple as porridge and yet make you see the real thing, like the greatest haiku of them all probably is the one that goes 'The sparrow hops along the veranda, with wet feet.' By Shiki. You see the wet footprints like a vision in your mind and yet in those few words you also see all the rain that's been falling that day and almost smell the wet pine needles.

Only those who truly love and are truly strong can sustain their lives as a dream. You dwell in your own enchantment. Life throws stones at you, but your love and your dream change those stones into the flowers of discovery. Even if you lose, or are defeated by things, your triumph will always be exemplary. And if no one knows it, then there are places that do. People like you enrich the dreams of the world, and it is dreams that create history. People like you are unknowing transformers of things, protected by your own fairy-tale, by love.

Direct experience decisively answers the nagging questions inherent in faith. There are usually two phases of direct experience; peak experiences and plateau experiences.

Peak experiences are relatively brief, usually intense, often unbidden, and frequently life-changing. They are actually "peak experiences" into the transpersonal, supramental levels of one's own higher potentials. Psychic peak experiences are a glimpse into nature mysticism (gross-level oneness); subtle peak experiences are a glimpse into deity mysticism (subtle-level oneness); casual peak experiences are a glimpse into emptiness (casual-level oneness); and nondual peak experiences are a glimpse into One Taste. As Roger Walsh has pointed out, the higher the level of the peak experience, the rarer it is. . . .

Whereas peak experiences are usually of brief duration—a few minutes to a few hours—plateau experiences are more constant and enduring, verging on becoming a permanent adaptation. Whereas peak experiences can, and usually do, come spontaneously, in order to sustain them and turn them from a peak into a plateau—from a brief altered state into a more enduring trait—prolonged practice is required. Whereas almost anybody, at any time, at any age, can have a brief peak experience, I know of few bona fide cases of plateau experiences that did not involve years of sustained spiritual practice.

When I compare life to a dream I do not mean to denigrate it as some sort of meaningless fantasy.   Life is too wonderful to be called an "illusion" unless we whisper the word in amazement, as we might when witnessing the most astonishing magic trick.  What could be more magnificent than this glorious universe, in all its multifarious extravagance?  Its awesome vastness and delicate detail.  Its impersonal precision and intimate intensity.  Its harsh necessities and lush sensuality.  This dream of life is truly marvelous.

As an individual, with your death there will be an end of you. But your individuality is not your true and final being, indeed it is rather the mere expression of it; it is not the thing-in-itself but only the phenomenon presented in the form of time, and accordingly has both a beginning and an end. Your being in itself, on the contrary, knows neither time, nor beginning, nor end, nor the limits of a given individuality; hence no individuality can be without it, but it is there in each and all. So that, in the first sense, after death you become nothing; in the second, you are and remain everything. That is why I said that after death you would be all and nothing.

In later Advaita, as we have noted earlier, a distinction is made between two types or kinds of spiritual experience related to transcendental consciousness:  that of savikalpa samadhi, the borderline experience, as it were [...] and that of nirvikalpa samadhi, the pure experience of Reality.

Savikalpa samadhi means "determinate" spiritual experience.  In it an awareness of duality is absent, but unlike susupti, the deep-sleep state, the emphasis here is not so much on the absence of duality as it is on the presence of non-duality. Whereas in susupti the self is still a knowing subject, although there is nothing there as such to be known, in savikalpa samadhi the self is aware only of the presence of Reality.  In susupti all the phenomenal activities self are suspended in a kind of serene blank; in savikalpa samadhi they are concentrated on the Real. In susupti the self is still very much the jiva; in savikalpa samadhi the self is passing into the Atman.

Nirvikalpa samadhi is the consumation of the process. It is the pure "indeterminate" intuition of non-duality.  In savikalpa samadhi the Self is aware of Reality; in nirvikalpa samadhi it is Reality.