Rebirth (punàbhinibbatti or punabbhava) is idea that at death the mind re-establishes itself in a newly fertilised egg, thus animating a new body and continuing the individual. The Buddha taught that the force propelling the mind into a new body is craving (tanhà), specifically the craving for pleasure, for identity and the craving to live.
The ultimate purpose of all Buddhist practice and training is to eliminate craving and thus stop the process of rebirth. The Buddhist doctrine of rebirth differs from the Hindu teaching of reincarnation in two significant ways. According to Hinduism, (1) an eternal self or soul (àtman) passes from one body to another and (2) eventually merges with the World Soul (Brahman). The Buddha taught that (1) nothing is eternal, that the individual is a flowing, constantly changing process and (2) that liberation or Nibbana is the ceasing of this process.
People often ask, "If there is rebirth, why don't I remember my past lives?" Most of us cannot remember anything that happened to us before the age of 2 to 5, so how could we remember something several years or decades before that? Also, human birth is very rare:
"Monks, suppose that this great earth were totally covered with water, and a man were to toss a yoke with a single hole there. A wind from the east would push it west, a wind from the west would push it east. A wind from the north would push it south, a wind from the south would push it north. And suppose a blind sea-turtle were there. It would come to the surface once every one hundred years. Now what do you think: would that blind sea-turtle, coming to the surface once every one hundred years, stick his neck into the yoke with a single hole?"
"It would be a sheer coincidence, lord, that the blind sea-turtle, coming to the surface once every one hundred years, would stick his neck into the yoke with a single hole."
"It's likewise a sheer coincidence that one obtains the human state. " Samyutta Nikaya 56.48
The blind sea turtle is a good simile showing how hard it is to obtain human rebirth. A goldfish only has a memory of about 3 seconds. Most other animals don't have that long of a memory either (except for elephants). If we have done all / most of our past lives as animals, we cannot expect to remember them much or at all.
Rebirth in the Suttas
The suttas (discourses) are permeated throughout with discussion of rebirth and what leads to the end of rebirth.
"He recalls to mind his various temporary states in days gone by; one birth, or two or three or four or five births, 10 or 20, 30 or 50, a 100 or a 1,000 or a 100,000 births, through many cycles of cosmic contraction and cosmic expansion . . .
. . . he recollects his numerous past lives: that is, (he recollects) one birth, two, three, four, or five births; ten, twenty, thirty, forty, or fifty births; a hundred, a thousand, or a hundred thousand births; many hundreds of births, many thousands of births, many hundreds of thousands of births. (He recalls:) 'Then I had such a name, belonged to such a clan, had such an appearance; such was my food, such my experience of pleasure and pain, such my span of life. Passing away thence, I re-arose there. There too I had such a name, belonged to such a clan, had such an appearance; such was my food, such my experience of pleasure and pain, such my span of life. Passing away thence, I re-arose here.' Thus he recollects his numerous past lives in their modes and their details." Digha Nikaya 1
Acceptance of Rebirth
There are some people interested in Buddhism who are known as skeptical Buddhists or agnostic Buddhists who have doubts about rebirth and some other teachings. This is considered acceptable as it is a "come and see" for yourself religion. However, some take their doubt to an extreme and prefer a Buddhism without rebirth and argue that concepts like rebirth could keep some away from Buddhism, believing it to be a cultural accretion. However, is reincarnation / rebirth really a hindrance to getting more appeal for Buddhism? Studies show that belief / acceptance of reincarnation / rebirth are at very high numbers. Here are some statistics of the percentage believing in reincarnation / rebirth: 
- Nordic countries: 22%
- Lithuania: 44%
- Western Europe: 22%
- United States: 27%
The above are very high percentages when you consider that the above countries / regions are primarily Christian (at least by birth certificate), which does not traditionally accept rebirth. The U.S. study showed that acceptance was especially higher among the younger age groups. Belief in reincarnation is held by 40 percent of people aged 25 to 29 but only 14 percent of people aged 65 and over.
- Erlendur Haraldsson. "[https://notendur.hi.is//~erlendur/english/Nordic_Psychology_erlhar06.pdf Popular psychology, belief in life after death and reincarnation in the Nordic countries, Western and Eastern Europe]".
- wnd. "27% believe in reincarnation".