The Indian Tradition
Hinduism and Buddhism share beliefs in reincarnation and karma. Karma is a cosmic law of moral cause-and-effect: your current station in life is determined by your goodness or badness in past lives; your current goodness or badness will determine who you become in your next life. "What goes around comes around." You can break out of the cycle by achieving nirvana, a state interpreted as enlightenment, ecstasy, or self-annihilation.
Hinduism
Hindus believe in karma and in the sacredness of the Vedas. Aside from this, however, there is enormous latitude. Many Hindus believe: (1) that the caste system is religiously legitimated; (2) that there is a variety of gods and god-incarnations, e.g. Brahma the creator, Vishnu the preserver, Shiva the destroyer, Krishna; and (3) that the individual soul, the "atman", is identical to cosmic reality, the "brahman". Yoga is the physical/mental /spiritual discipline by which one tries to yoke the atman to the brahman.
Extent: the dominant religion of India.
Sacred texts: the Vedas (includes the Upanishads)
the Mahabharata (includes the Bhagavad-Gita)
Buddhism
The Four Noble Truths: (1) existence is suffering ("there are more tears on earth than water in the ocean"); (2) the cause of suffering is desire; (3) one may transcend suffering by achieving nirvana; and (4) the way to do this is by the "Eightfold Noble Path".
The Eightfold Noble Path: right understanding, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, and right concentration.
Extent: Before dying out in its native home of India during the Middle Ages, Buddhism spread through East Asia where it now flourishes.
Sects include Theravada, Mahayana, Lama, Tantric, and Zen. Zen emphasizes the importance of meditation, for instance meditation over paradoxical koans -- "What is the sound of one hand clapping?"
Sacred texts: the Tipitaka (Kaviratna edition) / (Carus edition); commentary including Bardo Thödol and the tantric Kama Sutra.