Buddhism, a mass of teachings on living the good life that is popularly considered a religion, is in many ways non-theistic. The Buddha did not preach the existence of a deity, and is not worshiped, but rather, is paid respect by Buddhists. Although Buddhism doesn't necessarily recognize the supernatural at all, as it spread and assimilated with different local beliefs, some sects formed which integrate spirits, superstitions and other supernatural figures alongside the Buddha's teachings.
The main point Buddhism makes is that the way to achieve enlightenment, and to leave suffering behind entirely (these are essentially one in the same thing), lies in letting go of craving (wanting to 'posses' or be near good things) and aversion (wanting to get away from bad things). The aim is to replace these tumultuous feelings with compassion for oneself (not to be confused with self-pity, which is not realistic or healthy), and compassion and empathy for others.
The Buddha was a man said to have been born as a prince some time between 624 and 368 BCE. He spent his early years shielded from suffering by his royal father. One day while exploring outside the walls of his home, he came upon an old man, a sick man, and a dying man, and was overwhelmed with disappointment and shock. He left his wife and child to find a way to make sense of these negative aspects of life, and to learn how happiness might still be obtainable.
Since he had already tried living a life of luxury, he went the opposite route, and joined the ascetics - men who routinely deprived themselves of food, water, and even simple possessions in order to find enlightenment. He lived in a cave, he meditated for long hours, he starved and dehydrated himself, and finally, he realized he was making no headway. Unsatisfied, he left asceticism too.
He spent a night meditating under a bodhi tree. A specimen of this tree (the species is actually named after this story; 'bodhi' means 'awakening'), still thriving in India, is said to be the very tree under which the Buddha found enlightenment.
That night, the Buddha discovered the Four Noble Truths:
1) Suffering exists.
2) Suffering is caused by desire (craving and aversion; all negative emotion, and thus all suffering, arise from these things).
3) The end of suffering lies in finding enlightenment, called Nirvana.
4) There is a way to find the end of suffering - the Noble Eightfold Path.
He is said to have actually become the Buddha upon discovering this and obtaining the determination to help others understand it too.
The Buddha returned to his ascetic friends, and taught them about the Middle Way (a 'not too much, not too little' philosophy, similar to Aristotle's Mean), which included teaching them about the Eightfold Path, as directed by the Four Noble Truths. This was his first sermon, and the ascetics became his followers. The eight steps of practicing the Eightfold Path are as follows:
1) Right Speech (speaking accurately and truthfully)
2) Right Actions (avoiding action that would do harm)
3) Right Livelihood (choosing a livelihood that doesn't harm others)
4) Right Effort (always striving to improve)
5) Right Mindfulness (seeing the way things are; being conscious and clear-headed)
6) Right Meditation (being aware of the present moment, free from craving and aversion)
7) Right Understanding (understanding what is, beyond what appears to be)
8) Right Thoughts (an effective pattern of thinking)
The Eightfold Path is often symbolized by an eight-spoked wheel, which looks similar to that used to steer ships.
Other important Buddhist concepts:
The Three Jewels
The Buddha, The Dharma (the Buddha's laws/teachings), and The Sangha (the institution consisting of people who have at least started on the path to enlightenment), constitute the three jewels. This is where a Buddhist is to take refuge; the three jewels are in place to help a Buddhist along his or her path.
In Buddhism, karma is amoral. It is simply the complex system of cause and effect which affects everything from an ant's footsteps, to what happens in day-to-day human life, to the structure of the entire universe. Buddhism's main reason for studying karma is to learn its role in human affairs. Ideas like 'bad' and 'good' karma are really just applications of our own tendencies to avoid or crave certain things. 'Good' and 'bad' are not parts of the external world, but parts of ourselves.
"Atman" means "soul;" "anatman" means "no soul." The Buddha taught that humans do not have souls, which stood out in contrast to the popular Hindu beliefs of the area and time period.
Rebirth is often confused with reincarnation. Hindus believe in reincarnation with varying details (Hinduism is very eclectic), but the idea of reincarnation always requires the existence of a soul.
Rebirth, on the other hand, is closely related to karma, and is more complex and subtle than the idea of a soul traveling from body to body. The soul does not exist, but karmic patterns do and patterns can occur over and over again like ripples and waves and, shockingly, people and animals too. The theory of evolution fits nicely with Buddhist perspectives birth itself is a form rebirth, with nothing supernatural required. So is blinking your eye, and so are the thoughts in your mind.
The Absence of "I"
Not only did the Buddha teach that there is no soul, he taught that there isn't, in fact, a 'self' at all. That sense of "I," of "me," of "being something that it is like to be," - this is a very persistent illusion. It's what gives us the assumption that the soul exists. We don't remember a time when this "me" did not exist, and so it seems only natural to us that "I" will never stop, never die; that "I" will continue on after death. The idea that death is final is terrifying, because it's so unfamiliar and hard to imagine; and this fear is brought about by the very concept of "I" itself. Buddhism strives to relieve us of this sense of "I" while we are alive, as clinging to it is the most basic reason for our craving- and aversion-centered troubles. We often experience a loss of this "I" sense, or ego, without fully realizing it, even when conscious (not just when sleeping). For example, we lose ourselves when absorbed in a task, or when captivated by a story. Buddhism seeks to deepen these experiences and apply them to the way we live.
The Buddha went on to teach until he was in his eighties, and many stories are written about his travels. He is said to have died in an interesting way: by the poisoned offerings of a gracious host. He knew the food to be poisonous, but also knew that the host had good intentions, so he consumed the food so as not to be rude, while directing his followers not to touch it. It was all right that he would die, he said - he had finally fully found Nirvana, and his followers could carry on his teachings in his stead.
Buddhism mainly spread eastward from India, and is now the fourth most popular religion in the world, with an estimated 325 million followers. It branched off into two main schools.
Theravada ("Little Vehicle") Buddhism has many followers in Sri Lanka and Southwest Asia. It is conservative, and is also the oldest school. It places great emphasis on adhering to scripture.
Mahayana ("Great Vehicle") Buddhism is more inclusive and eclectic, with followers mainly in China, Korea, Tibet and Japan. It is very adaptable and inclusive, having integrated the cultures of the regions it spread to like so much material on a Katamari Damaci ball. Zen (very abstract and, to some, cold), Shinto (native to Japan, Shinto amalgamated Buddhism with many local legends) and Pure Land (which contains a concept of a heaven-like place) are three of Mahayana's sub-sects.
Daoism (or Taoism) takes a simpler approach to life, and is a good friend of Buddhism, but doesn't quite fit under Buddhism's umbrella. It is based closely around I Ching (or book of changes') which is a partially-supernatural divination practice studying karmic cause-and-effect.