Many people associate Buddhism with reincarnation, but in fact, the Buddha taught the concept of rebirth, which is slightly different. Reincarnation implies the migration of a soul, an intrinsic essential self, from one physical form to another. Buddhism teaches that there is no irreducible self, but instead a constantly changing set of mental and physical components called the five skandhas.
These skandhas are constantly in motion, separating and combining into different forms, including what we think of as our bodies, personality, thoughts, and emotions. Our clinging to these skandhas, and our belief that they add up to an impermeable self or soul, is the root of our suffering, or dukkha, one of the three marks of existence in Buddhism. This clinging is caused by avidya, or ignorance, and Buddhist teachings and practices are designed to help us break free of this ignorance.
With his teachings on rebirth, the Buddha was responding to and refuting certain aspects of Hindu teachings on reincarnation - the teachings he was raised with. His teachings were based on his direct experience of impermanence, or anicca, also one of the three marks of existence, which are themselves central to understanding the foundation Buddhist teachings of the Four Noble Truths. In English translations of Eastern texts, the terms 'rebirth' and 'reincarnation' are often used interchangeably, which has added to confusion on the difference between the two.
A metaphor often used to explain rebirth is that of lighting a candle with the flame of another candle. Although there is a relationship between the two flames, they cannot be said to be the same, nor are they entirely different from each other. In the same way, our consciousness in this lifetime is related to, but neither the same nor different from, consciousness in prior lives.
Buddhist meditation offers a way to understand this idea of rebirth in the context of every mind-state that we experience. Each perception, sensation, emotion, or thought that we experience arises in our consciousness, holds our attention for awhile, and then passes away. Subsequent mind-states are related to, but neither the same nor different from, our prior states. If we drop our idea that a continuous self connects each state, we can experience every moment as the birth of a new consciousness.
The different schools of Buddhism vary in their interpretation of how rebirth functions. Most Theravada schools teach that rebirth is immediate, while many Mahayana schools, including Tibetan Buddhism, teach that there is some intermediate state between lives, or bardo. The Tibetan Book of the Dead elucidates one view of these intermediate states in detail, and provides instruction for navigating through them to influence future lives or even to attain nirvana.
Another variation in the Buddhist theories of rebirth relates to the idea of tulkus, or Tibetan Buddhist lamas that have chosen to be reborn in order to continue teaching - of which the Dalai Lama is one. These tulkus are considered enlightened bodhisattvas, liberated beings that are no longer bound by ignorance to skandhas, and therefore are no longer subject to the usual rebirth process. Instead, they choose to be reborn out of compassion for humanity, in order to aid other beings on their pathway to liberation.
The different schools of Buddhism also vary on the level of importance they place on rebirth teachings. As Buddhist teachings have evolved in the West, some teachers choose not to emphasize it with their students, realizing that it is an unfamiliar concept. Most teachings also emphasize that it can become a distraction - focusing on who we were, what we did, or where we lived in a prior life may just serve to reinforce our sense of ourselves as a continuous self, strengthening our attachment to ignorance. On the other hand, recognizing the constantly changing state of our own awareness through Buddhist meditation and mindfulness may naturally lead us to an understanding of rebirth as the Buddha taught it, and this latter approach is the real purpose to contemplating rebirth as part of a Buddhist practice.
Note that this article is included in my e-book Introduction to Buddhism and Buddhist Meditation.