The principle of not-self is not saying "No self", it's saying a specific kind of thing:
"Bhikkhus, feeling is not-self...
"Bhikkhus, perception is not-self...
Upon the greater question of "does some kind of self other than the aggregates exist", the Buddha refused to answer:
"Malunkyaputta, did I ever say to you, 'Come, Malunkyaputta, live the holy life under me, and I will declare to you that... 'The soul & the body are the same,' or 'The soul is one thing and the body another,'...
"And did you ever say to me, 'Lord, I will live the holy life under the Blessed One and [in return] he will declare to me that... 'The soul & the body are the same,' or 'The soul is one thing and the body another,'...
"So, Malunkyaputta, remember what is undeclared by me as undeclared, and what is declared by me as declared.
And what is undeclared by me?... 'The soul is one thing and the body another'
He says "when there is the view, 'The soul is one thing and the body another,' there is still the birth, there is the aging, there is the death, there is the sorrow, lamentation, pain, despair, & distress whose destruction I make known right in the here & now."
When most people think of "self" they think of some abstract core that is the subject of all experience and the agent of all actions. Buddha taught (and modern cognitive science tends to agree) that upon careful examination there is no such single core. Instead experiences result from interactions of multiple perceptory functions. Similarly, our actions are performed by a conglomerate of functions, without a single agent responsible for all choices.
When the notion of individual rebirth exists as an unexamined belief, it depends on this unexamined notion of "self" as its necessary foundation. Once practitioner is free from illusion of self (in practice, not just conceptually), the notion of individual birth/death/rebirth no longer applies. From this perspective, rebirth is a byproduct of attachment to a (substantial) self.
But because karma at large still continues to function, the infinitely forking/joining threads of causation that once were the subject of attachment continue to participate in activity and serve as causes for subsequent effects. In this sense, rebirth as principle goes on continuously whether we posit an individual or not.
To give this a more practical spin, instead of worrying about rebirth, we practitioners should focus on working towards deconstruction of self by reducing the strength of the illusion through practice of altruism and non-attachment, first and foremost at the microscopic level of thought-to-thought intention and reaction.
This is an excellent question. Karma is well accepted in Buddhism, but not the existence of the soul. This inconsistency raises the question of remnant reaction. Consider the following argument.
From the system of karma we know that a person who does the action will be the same person who will experience the reaction, not any one else. Suppose a person dies, then what will happen to the inexperienced remnant reactions? It can't just go away, can it? No, and we know from system of karma that the same person will born again to experience the pending reactions. So here is the question, what is common in both the persons that you say the new born one is the rebirth of the old deceased one? If the two persons are identical then there has to be something common in them, isn't it? So what is that fundamental unit that is common to both? It can't be the body, nor can it be the mind (as memory, tendencies, qualities, etc. are different). So what is the common element? The answer is soul. I have to keep it brief so can not go elaborate more but the short answer is soul exists.
Why did not Buddha acknowledge the existence of soul?
The mission and objective of Buddha was to ward of suffering of the people which is primarily caused by attachment. So He didn't acknowledge the existence of soul. But just because He didn't acknowledge its existence, it doesn't necessarily mean it doesn't exist. He didn't acknowledge the existence of soul because it will create attachment to an eternal self. And where there is attachment suffering will undoubtedly follow. During His time in India people were doing Vedic sacrifices, yanjya, rituals, etc. to place there self in heaven etc. As they believed in the eternal soul, they were attached to it and hence putting effort to place it in heaven and higher realms by animal sacrifices in yanjya and rituals. So He didn't acknowledge the existence of soul to free followers from attachment towards an eternal self.
However, if you go into more deep then you will find that the soul exists in such a way that its existence can be safely overlooked. So even if you deny the existence of the individual soul you will not be completely wrong.
Anatta is often described as "not-self" which I understand to mean that our identities are illusions.
No, in fact, the meaning of not-self, as others have pointed out, is that the object in question is not self :) The Visuddhimagga offers some good explanations:
All that [materiality] is “not-self in the sense of having no core.” In the sense of having no core because of the absence of any core of self conceived as a self, an abider, a doer, an experiencer, one who is his own master; for what is impermanent is painful (S III 82), and it is impossible to escape the impermanence, or the rise and fall and oppression, of self, so how could it have the state of a doer, and so on? Hence it is said, “Bhikkhus, were materiality self, it would not lead to affliction” (S III 66), and so on. So this is also not-self in the sense of having no core. Accordingly, there is one kind of comprehension in this way too, but it is effected in eleven ways.  The same method applies to feeling, and so on.
There are twenty-five kinds of contemplation of not-self by taking the following five in the case of each aggregate: as alien, as empty, as vain, as void, as not-self.
But it's also described as "soullessness" which I think implies that there is no mind other than the brain itself.
Absolutely not! The five aggregates are non-self, four of which are outside of the domain of the brain.
If there is no soul, how can there be rebirth?
There isn't in fact any such thing as rebirth, in ultimate reality. Nothing is truly reborn, ever. Every experience that is made up of the five aggregates arises and ceases without remainder. That is true death. Rebirth is a concept used to describe the change between one artificial framework of experiences (e.g. a human life) to another. The ultimate reality is that the mind simply arises and ceases at the last moment of life and then a new mind arises at the first moment of rebirth based on the last one, very similar to as has been occurring throughout one's life, except this time there is no old physical phenomena for it to be based on, so it is based solely on one's final state of mind in the last life.
Here's something that might help:
Only "eternal, unchangeable, and happy" Soul (which is called atman) is denied in Buddhism. And such monistic Soul is not required for re-birth to function. When being disintegrate it is causally condition birth of new being. Thus, even though there is no constant Soul, two beings are related to each other causally by means of karma. So, instead of Soul (and re-incarnation) we have Karma (and re-birth).
All most all above answers are helpful in getting rid of confusions regarding rebirth. Still I will say something in this regard as I think it will be an answer in quite simple words.
I am pasting my answer to other question regarding rebirth. That question also has another link. This link will take you to the original text based on which I have tried to answer.
Hopefully that answer is suitable and in very 'easy words'. -
According to the Buddha, there are four elements of Existence which go to compose the body. They are (1) Prithvi; (2) Apa; (3) Tej; and (4) Vayu. Question is, when the human body dies, what happens to these four elements? Do they also die along with dead body? According to The Buddha they join the mass of similar elements floating in (Akash) space. When the four elements from this floating mass join together, a new birth takes place. This is what the Buddha meant by rebirth, as I have understood it. It must be noted that the body dies. But the elements are ever-living. This is the kind of rebirth in which the Buddha believed. Great light is thrown upon the subject by Sariputta in his dialogue with Maha-Kotthita. The Buddha was not an absolute annihilationist. He was an annihilationist so far as soul was concerned. He was not an annihilationist so far as matter was concerned. (I think Andrei Volkov is also saying the same point when he is talking about "complete annihilation).
There is another concern here. It is about rebirth of Whom? Does the same dead person take a new birth? Did the Buddha believe in this thesis? The answer is, "Most improbable." The answer depends upon the elements of existence of the dead man meeting together and forming a new body; then the possibility of the rebirth of the same sentient being is possible. If a new body is formed after a mixture or the different elements of the different men who are dead, then there is rebirth but not the rebirth of the same sentient being. This point has been well explained by sister Khema to King Pasenadi. Where she ends up saying 'The Tathagata exists after death...exists not after death', does not apply." (Again we can match Jayantha's write up regarding Kalama Sutta.) I have drawn lot from The Buddha and His Dhamma Book 4, Part 2, section 1, sub section 2. Hoping for further discussion.
The "answer" seems to be more like a glass of water being taken from a river and then being poured back into the river. Can the exact same glass of water ever be drawn from that river? Possibly, but extremely improbable, but another glass of water may be drawn from the river, a river that has been influenced by the actions of the first glass of water. Analogies never take in all angles but I like this one.
Seed has a tendency to produce a life again. So seed contain the complex that creates new life. But when the same seed has been burnt and popped, will it return to seed again and create anew? No.
So whatever there is the condition to reappear, has the tendency of cessation. Like popping corn.
Hence if you take seed as your self, creation continues. If you burn that seed and make into pop corn, will you still think you have soul? No.
Hence all things are impermanent, interdependent, cause of one another and no self!
Hope you will burn that seed!
There is no rebirth / reincarnation (in the sense that the soul takes on a new body after the death of the previous body. )
Our existence is changing process in which there are events like birth and death. In the absence of a unchanging everlasting core the person who is born is not the same as the person who died nor is he a deferent person but but just the continuation of the process of existence.
In Buddhism what is there is Jati / birth and death as events in the process keeping us in the cycle of Samsara.
There is no continuity of the person (or some core identity / entity) but there is a continuation of the process.
This is a question of kings! Many many years ago, a king asked the same question of a monk. Here is the dialogue from Sutta Central
The king said: ‘Where there is no transmigration, Nāgasena, can there be rebirth?’
‘Yes, there can.’
‘But how can that be? Give me an illustration.’
‘Suppose a man, O king, were to light a lamp from another lamp, can it be said that the one transmigrates from, or to, the other?’
‘Just so, great king, is rebirth without transmigration.’
‘Give me a further illustration.’
‘Do you recollect, great king, having learnt, when you were a boy, some verse or other from your teacher?’
‘Yes, I recollect that.’
‘Well then, did that verse transmigrate from your teacher?’
‘Just so, great king, is rebirth without transmigration.’
‘Very good, Nāgasena!’
Those are metaphors, but what actually passes from life to life? Habits! All we have to do is break the habit of wrong desire. Here is another reference: sutta central where habits of wrong desire are compared to feeding the oil-lamp.
I am not a Buddhist, but in reading "the Essence of Buddhism" by P. Lakshmi Nasaru, what strikes me is how the Buddha wants his followers not to accept blindly anything, but to question everything and come to realize their own conclusions. Independent thought and deep inquiry in oneself, seems to be the way of progress, not acceptance of authority. Perhaps this is the final test: after purifying consciousness through all the practices, over a long time, you finally realize you ARE an eternal soul, and then as a result, in the end, you must give up attachment to the Buddhist doctrine. Tricky, this Buddha.
First, you need to understand that Buddhist texts operate on two levels of description: the conventional truth and the ultimate truth. The contradiction disappears when you realise that anatta (the non-existence of self) is an ultimate truth, while the rebirth is a conventional truth.
In other words, Buddhists believe that people linked by a chain of rebirths may be said to be "the same", by the same convention as when a 40-year-old person is said to be "the same" as an 20-years-old person he evolved from. This convention has nothing to do with the fact that there is no indivisible and constant "I", and that the aggregates that make up the conventional "I" are in constant flux.
Think of exchanging the terms. If there is no rebirth how can there be a soul. Attachment will be there both ways. Why must one condition rely on another. Why must a soul be present for a rebirth ? Why must there be a rebirth to validate a soul. Soul is a category based on a disposition masking itself as a representation. You are not a soul. You are not a rebirth. You are not a name. You are not a species. You are not a category. You are not a self. You are not a hope to be. You are not a never was. You are a system of directed energies subjected to and affected by other directed energies. All religions and convictions are merely reminders and reinforcers of how we are to be. Like signs on the road. How do you direct your energy ? Will believing you have a soul help you in redirecting your energy ? I do not know this .
It's like saying there is not-self or soul so consciousness cant move forward. Consciousness throughout life, throughout Samsara, is always dying in every fraction of a second then arising again in the next. When consciousness arises again that is a rebirth.
Why do we assume that conciousness won't continue after physical death? Mindfulness meditation challenges those kinds of assumptions.
The Pali word 'atta' is generally translated as 'self, body, person, individuality; life, mind-soul.' As the PED makes clear, the Buddha is alluding to a very specific concept - the Upanishadic idea of a substantial being that inhabits the heart that is reborn from life to life and is inherently permanent, self-identical, and blissful. The Buddha points out that no amount of observation can confirm the existence of such a being. Instead, he posits a continuity of moments, each moment infintesimal and non-substantial, which he calls the 'santana,' generally translated as 'continuity, succession,' often referred to as the 'mind-stream.' The difference between the 'atta' and the 'santana' is that the former is substantial and static, whereas the latter is non-substantial and dynamic. The 'santana' manifests in samsara as a consecution of states resulting from unfruited karma, which in turn arises because of objectification due to ignorance (avijja). However, when all karmas are eliminated there is "that which remains," which continues to be non-substantial and dynamic. This is the Deathless (amara, same root as amrita). The Buddha clearly rejected nihilism, he refers often to the attainment of immortality and refers to the "self" as frequently as he denies its existence. The notion that the Buddha denied the existence of a self in any sense is naive, and is easily disproved. When the Buddha attained Enlightenment, only the Buddha became enlightened, nor did the Buddha himself disappear. This proves the existence of the individuality, the essence of which is intention (cetana), the primary characteristic of which is 'not-knowing' (avijja). This is a primary aspect of reality itself, one polarity of the transdual absolute of which the opposite polarity is emptiness or void (shunyata). What the Buddha rejected is the doctrine of a being inhabiting the heart with the characteristics described. In other words, he is opposing the objectification of the santana as a static object. It is the process view of reality (cf. Whitehead's Process and Reality). Esoterically, we may say that the ATTA is the A-TA, the "not-that." See also the Dalai Lama, Buddha Nature.
In original Pali Buddhism, there are two levels of teaching: (i) mundane, for the purpose of nurturing morality; and (ii) supramundane, for the purpose of enlightenment (MN 117; MN 60). The teachings that imply 'rebirth' are mundane & the teachings that explain 'anatta' are supramundane. If we carefully read the many Pali suttas, we will not find 'rebirth' & 'anatta' in the same sutta. They are two distinct topics for two distinct audiences.
For example, MN 60 states the view of 'self' or 'existence' is right view for householders but SN 12.15 states the view of 'self' or 'existence' is not right view.
The word 'birth' ('jati') refers to the birth of the idea of 'self-identity' ('sakkaya' - MN 44) or 'persons/beings' ('satta' - SN 12.2; SN 23.2; SN 5.10; SN 22.81). Minds that are not enlightened are subject to 'rebirth' of the deluded 'self-view' each time they cling to an act of karma. Minds that are enlightened have ended 'birth' & have ended clinging karma (AN 6.63).
Because the old Pali suttas do not mention anything that is 'reborn' from life to life, the later Buddhists were required to invent what they call 'relinking consciousness', 'storehouse consciousness', 'stream of consciousness', etc, which are ideas the original suttas did not teach. The original suttas only state consciousness is the six-fold cognition (MN 43) and there can be no arising of consciousness without sense organs (MN 38) and aggregates (SN 22.53; SN 12.67).
If there is no soul, how can there be rebirth?
I think this is a very important question and one which really gets to the heart of a big misconception about emptiness or the selflessness of persons.
To my mind, the best way to see the hidden assumption in the question above is to ask another question: why invoke rebirth?
How about this alternate question: If there is no soul, how can there be a person who lives moment to moment?
It is clear, that all schools of Buddhism teach the selflessness of persons. This is one of the four seals of Buddhism. So, why don't we ask how can there be a person who lives moment to moment?
It's manifestly evident that people live moment to moment. With just a little bit of insight we can see that persons are impermanent. That we are non-unitary (we have parts). That we are dependent phenomena. That right there is enough to establish the selflessness of persons at a course level and yet we still function and live moment to moment.
Rebirth is simply the continuation of this living moment to moment even after the death of the body. In other words, the question has two hidden assumptions:
The Dharma does not contain these assumptions and there are good reasons for rejecting both.
If there is no rebirth how will you come back to find out???
Your understanding of “soullessness” needs attention. This is wrongview of annatta i guess