Hinduism is not technically one religion at all, at least not in the same way that Roman Catholicism is one religion, but the loose association of a myriad different yet similar belief systems found priamarily in and around the Indian subcontinent. Hinduism lacks a known founder, with an unclear history that goes as far back as 4000 BC, depending on who you ask, and to this day it has no central leadership to speak of.
The fundamental idea governing the Hindu worldview is the concept of brahman, the essence of the universe. Atman, the self or the individual, is actually equivalent to brahman. (Be sure not to confuse brahman with Brahma the god, or with my username!) So, each individual is a part of the whole. It is easy to see how a version of the Golden Rule manifests itself here, since from a Hindu perspective, anything you do to another you do directly to yourself, since you are him and he is you.
In concert with this central unity exists a vast array of gods and goddesses, with some counts numbering them in the neighborhood of 300 million! How can this be, if everything is one? I think perhaps a good comparison is with the Christian Trinity. The many deities can be understood as different aspects of the central oneness, or however else you would like really, since Hinduism tends to be inclusive of different, even seemingly contradictory interpretations, as opposed to exclusionary.
The three main gods of Hinduism are Brahma, the creator; Vishnu, the sustainer; and Shiva, the destroyer. These three are associated clearly with the wheel of rebirth known as samsara. Most Hindus focus their worship primarily on either Vishnu or Shiva. Another popular god is Ganesha (you know, with the elephant head). Also significant are avatars, or incarnations, of the gods, primarily Krishna, an incarnation of Vishnu. The conversation between Krishna and Arjuna forms the basis for one of the most popular Hindu sacred texts, the Bhagavad Gita.
It is often said that a central tenet of Hinduism is reincarnation. This can be true. The atman, with associated karma, is said to leave a body as it dies and inhabit the body of a newborn (newly conceived, whatever) baby. (Note that this baby is not neccessarily human.) Some would say that this is no more reincarnation than is the movement of a soul from your body now to your body a second from now. It is not the same body, really, and yet you find continuity of person in this slighty different body. In a longer time frame, your body now is as dissimilar to your body at birth as it is to that of any other human baby.
The body your atman inhabits in your "next life" is determined by your karma. Similar to the karma of slashdot, it records how good or bad you have been, so to speak. It is determined by your actions in relation to your dharma. Dharma, in turn, is the rules of how you should act and what you should do. It is determined by your age, your sex, and your caste. The caste system of India, which was made officially illegal around 1950, is based on the concept that you are born to a life that you deserve. There are four major divisions of caste, plus the immediately recognizable untouchables.
And of course, no writeup on Hinduism would be complete without a reference to Gandhi!