Buddhism grew out of Hinduism (much as Christianity grew from Judaism) around 300 BC. Buddhists believe in the cycle of birth, death, and rebirth of the Hindus. The difference is that the Buddha (in life an Indian prince) showed how to break out of the infinite cycle. Buddhists look at the Buddha as their guide to Nirvana, the state of enlightenment reached when one breaks out of the cycle of life.
Buddhist believe in moderation in everything, the so-called middle way. It is a mellow gentle religion which is usually tolerant of other faiths. Many Buddhists are vegetarian abhorring the taking of life. Their main festival is the Thrice Blessed Day on which the Buddha was born, reached enlightenment, and died (in different years, of course).
The holiest sites are in northern part of the Indian Subcontinent. Lopburi, in Nepal where the Buddha was born. Bodh Gaya where he reached enlightenment (the story goes that he was sitting under a Bo tree). Sarnath, the place where he gave his first sermon. The latter is a deer park because the story says that even the deer looked around when he preached.
In Buddhist countries most men and many women become monks even for a short time. Monks dress in orange and collect food from the public one a day. They are a common sight in the morning in even the most modern Buddhist cities. Buddhists spend a lot of time chanting their scriptures (these chants are called mantras). They are written in Pali, a derivative of Sanskrit and forerunner of Sinhalese.
Buddhism spread from India to Sri Lanka, Tibet (and thence to China and Japan), but is strongest in South-East Asian countries like Thailand, Burma, and Cambodia.