If I may add my own insights and personal understandings to this thread:
Hindus believe in reincarnation, transmigration of soul, rebirth, or whatever it's called these days. The basic premise is that something living inside you [soul/atma] leaves your body and enters a new body, and is reborn life after life. So we end up with this strange picture of all these millions of souls on a merry-go-round that leads nowhere but round in a circle between mortal earthly lives.
The Buddha did not seem to like this idea. He seemed to believe that the idea of a soul was reified. So he countered the Doctrine of Atma [Atma: that which is reborn] with the insight of Anatta [No-Atma]. Basically in very simple terms the Buddha with the doctrine of Anatta directly opposed Brahmanism's belief that an Atma existed in the first place to even reincarnate. This was a very bold move, because in Buddha's days, Brahmanism was the reigning religion. It would be like a person challenging the Church's doctrines and teachings in the 1400's in Europe you see.
In place of reincarnation [as the meme is used and understood in Brahmanism and the Upanishadic Traditions] the Buddha taught two interconnected insights. I use the word "Insight" as opposed to "doctrine" in respect to what the Buddha taught because the word Doctrine has a feel that if you challenge it, you are a heretic, when the Buddha himself reminded us to use our own thinking to figure things out and to even challenge what he taught. Enlightenment does come from following and believing blindly in another persons teachings. Following and believing in another persons ideas actually goes against the concept of Sama Sambuddhi: The Self Enlightening Itself; or One Enlightening Oneself.
The two Insights the Buddha gave are not commonly studied by Buddhists. And when they are known and understood, the two are rarely put together. It's when you learn to put the two Insights together that you begin to see the bigger picture.
The first Insight directly deals with Samsara and is called the Bhavachakka [Bhavachakra in Sanskrit]: The Wheel of Becoming. Samsara in Theravada Buddhism happened in the Bhavachakka, hence the idea of a Cycle of Life, due to the fact that the Bhavachakka is apprehended as a Wheel.
The Bhavachakka has 4 "planes of existence" -
1) Apaya Bhumi
2) Kamasugati Bhumi
3) Rupavachara Bhumi, &
4) Arupavachara Bhumi
Each of these four planes have "realms" or divisions or "worlds" of their own. All together there are about 32 Realms or Planes of Existence in the Bhavachakka. So Samsara happens Bhavachakka-wide.
Depending on your state of being, your "karma," and your level of Chitta/consciousness, when you/we die we "go" to anyone of these 32 places; and any living being in any of these 32 places are also subject to Samsara, meaning that when they "die" they will end up in different places in the Bhavachakka.
So in Buddhism Samsara does not suggest some eternal cycle of birth and death where the soul is trapped on the Earth for thousands of lifetimes. That's a pretty pathetic cosmological model when you have this infinite universe/cosmos but all of its life forms are stuck in some crazy cycle on some arbitrary planet out in the middle of nowhere Milky Way.
The second plane of the Bhavachakka is where our Loka [our physical universe/world] is found. Our realm of causal/physical existence is the lowest of 7 realms of the Kamasugati. The meaning of the Plane of existence we exist in adds insight to the fundamental essence of the Plane we exist in.
Kama as in "kama sutra" which is something we are familiar with. Kama meaning Sensual [having to do with the 5 senses] pleasure. Sugati is the verb form of Sukkha/Suga. Sukkha is the etymological opposite of Dukkha. Sukkha as a meme covers the semantic field of: "The state of being which is devoid of mental worry and trouble." So, put together Kamasugati means the Realm/Bhumi of Sense Delight and Happiness.
Thus, stimulation of the senses and happiness/tranquility [Sukkha] is the fundamental essence our Plane of Existence. If you are experiencing Dukkha [Unpleasantness, worry] then there is something wrong with the way you are living. Hence, the Buddha comes to teach us why we experience Dukkha, and gives us a way to destroy it, so that we may experience life the way it was intended to be.
Each of the 4 planes of existence are progressive in relation to each other with the Apaya Bhumi being the lowest and most primitive. The apaya bhumi - Realm of Misery - is where Dukkha and Papa is experience; not in Kamasugati Bhumi.
Each of the four planes are inhabited by beings [satta] that are likewise progressive in relation to each other.
Sattas - meaning Creature, Being, and "Animal" - exist in the apaya bhumi. Satta-Manussa - human creature/being - dwell in the kamasugati. Satta-Devatta in Form/Rupa exists in the Rupavachara. And Formless Satta-Devattas exist in the highest plane: the Arupavachara.
The word "Devatta" is sometimes mistranslated as "god," or "demi-god," which is not right. The root word DEV in Sanskrit and DEB in Pali means something "Bright," "Shiny," or "Luminescent."
So, these three classes of Sattas shows something peculiar. There exists some sort of evolution or progression of Chitta. Chitta [Consciousness] first begins in a primitive state of Satta [meaning creature]. It then becomes Manussa [human]. The root word in Manussa is MAN which means "To Think," and "Mind." From Manussa, Chitta evolves to become a Deb-Satta or Devatta - a Being of "Light."
This evolution of Chitta then takes us to the second Insight the Buddha taugh: Chitta-Samtana or sometimes called Citta-Santana; which in English is rendered: Mind-Stream, or the Flow of Consciousness.
The Buddha did not teach reincarnation, he taught citta-santana. He did use, and you/we will find in the Tipatakas usage of words like "reborn," and past lives, but the usage of these words is a matter of spacio-temporal context and convenience. People living in 500-400BC conditioned inside the memeplex of Brahmanism were already used to the ideation and concept of rebirth and "past lives." To be understood, the Buddha has to use terms the common market [the populous] were thinking in. "Common market" here is used in a marketing and advertising sense: you can't sell an idea to your target market, if the idea makes no sense. If you want to sell a new and innovative idea to the Brahamist Market, you need to dress up your innovative ideas with a coating of something familiar first. One must learn to apprehend Buddhism in certain respects to its native space-time context: that of India, 500BC, dominated by Brahmanism, not 21st century Earth.
Brahmanism's doctrine of Atma - that there is a soul/spirit - does not make any sense in the concept of citta-santana. I will try to explain.
If we took a 30 year old and magically brought out of this 30 year old his 29 year old self, his 28 year old self, his 27 year old self... all the way down to his primitive 1 year old self - and line up all 30 of those "peoples" we would see 30 different Selves/Atma. Each of this hypothetical guy's Selves is very different from the other in mental capability, though process, level of understanding, likes, dislikes, worldviews, beliefs, and so on. Some of these people are very primitive in Consciousness [the 1-5 year olds]; some are childish and are only interested in playing [the 5-12 year old]. Some seem to be only interested in sex and physical interests [the 13-18 year olds].
The question is: are all 30 of these very different "people" actually different beings/atmas or are we seeing/observing a single Consciousness/Chitta in its different grades/states of evolution/Becoming?
The other question is: If all 30 of these people are in fact the same Essence/Consciousness in different manifestations of becoming, then are We all different people... or just the same Essence/Chitta in different states of Becoming? When you try to answer this question, Nibbana/Nirvana/Moksha - the Realization of Liberation will make more sense. Nibbana sometimes is said to be the desolation of the Illusion of "Self," and the Realization of what you really are. It is the Awakening: Buddha.
Now we take that idea and superimpose it on the Bhavachakka, and ask ourselves if the very different beings - Satta, Manussa, and Devatta - are different species of beings, or are they consciousness in a state of Becoming/evolution [Bhava]?
The aim or objective is to escape the Bhavachakka to get to the "other side," which is what the Buddha came to do, so it is said.
People lost inside the Bhavachakka, mentally conditioned to understand only what they perceive and are born inside, become confused with the idea of reincarnation, rebirth, samsara, cycles, and being trapped. It's often asked why we just can't walk away from samsara in the first place?
The same people never stop to realize that while they are asleep and dreaming - while their Consciousness is immerses and entrances in the world of dreams - has no power to walk away from a dream and wake up at will. We are literally stuck inside our dreams until we wake up. If it weren't for the fact that we all naturally wake up in the morning, most of us would be stuck in a coma: unable to Awaken/Buddha.
How many people here have tried to actually Wake up in a dream? It's called Lucid Dreaming. It is possible, but it takes a lot of practice. I can sometimes do it. If it is that hard to learn to Wake up inside a dream; how harder is it to Awaken inside this dream we call Life? We are each stuck in a coma where our Chitta is engrossed with and lost in some cycle of Bhavackakka. The Buddha is like a doctor who tries to wake you/us up. This reminds me of the old move called the Cell and the Matrix, if anybody is familiar with it. Buddha is like Morpheus, and Bikkhus are like Neos. Remember Neo Awoke himself from his digital dream: Sama Sambuddhi - The Self-Awakened One.
I'm not sure if what I explained made any sense. I'm not saying that mortal existence is maya/illusion and fake. It's very real, but it is not the Supreme Real beyond Bhavachakka. We are in essence "trapped" in samsara as we are trapped inside our own dreams at night. We fail to realize that there is anything more to all of this because we were all born and conditioned only in this environment.
It's like old world Europe in the 1500's you see. If you were a person born and raised during that time and era, you would be raised to see and believe that the world is Europe. Your daily activities - what you attach and fixate your mind/chitta on - such as working, finding a mate, raising children or whatever keeps you "karmically" so occupied that you have very little time or interest in sailing a boat beyond the shores of Europe to explore. There is commotion in this hypothetical Europe where some people are now asking themselves if this Europe is the whole world, and if they are eternally stuck in a cycle of birth and death in it.
And then Christopher Columbus returns and says to everybody: "Hey, guys, I just found a entire New World, Wake up, theres more to all of this than Europe!" Buddha is like Columbus. Bhavachakka is like old world Europe. Nibbana is like the New World. Some of us, due to our misconceptions of what Nibbana is are reluctant to make the voyage. Like the old world people were afraid to make the voyage fearing they would fall of the edge of the world. Like many ill informed Buddhists fear that they will stop existing when they reach Nirvana. They never existed in the first place.
We only assume we exist and are "real." In the same sense that in dreams we believe and "know" that us and everyone and everything around us is real. But that whole dream world, and everybody in it is a manifestation of a single mind between your ears. When you/we Wake up in the morning, and our dream selves and those dream people fade into oblivion, we don't actually stop existing do we? Because who we believed our "selves" to be did not really exist in the first place.
This is one of the reasons why Buddha asked people or taught people to not be so attached to the world. To not fixate your chitta to deeply into the world and hold onto it obsessively. It's like a person in a coma who holds onto his dream form in his dream and refused to let go because he is so fixated in consciousness in his comatose world and comatose Atma/Atta. That grip, hold, obsession, attachment, fixation of one's own consciousness is what keeps you bound in Samsara. It is your/our own fault that we are "stuck" for as long as we are stuck. No body can Awaken you out of this coma but you yourself.
“Do not believe in anything because you have heard it. Do not believe in traditions because they have been handed down for many generations. Do not believe in anything because it is spoken and rumored by many. Do not believe in anything because it is found written in your religious books. Do not believe in anything merely on the authority of your teachers and elders. But after observing and analysis, when you find anything that agrees with reason and is conducive to the good and benefit of one and all then accept it and live up to it.” – Buddha (Anguttara Nikaya, Vol1, 188-193)