Hinduism
There is a paucity of information in the West as to what is the religion called "Hinduism". The truth of the matter is that it is no single religion, but a grouping of religions, all of which originated at different times and locals in what is commonly called the Indian Subcontinent.
Anthropologically speaking, the earliest two religions of the Bharat Peninsula (as the Indians themselves often refer to their country) were Vaishnavism, as epitomized by the various Vedas and Upanishads, and Shaivism, found in the earliest texts in Tamil.
The Vedic civilization is believed to have formed around the Saraswati River, now mostly a dry ditch seen only in satellite photos, which relocated to the area around the Indus River when the Saraswati dried up (in other words, modern day Eastern Iran and Afghanistan, relocating to modern Pakistan). The Vedas themselves may (or may not) predate the Saraswati civilization; indeed, some find passages in the Sama Veda which appears to have originated either in the farthest Arctic regions or even off-planet, depending on who you believe. The infamous caste system of present-day India was due to a mis-reading of the Vedas; originally it was little different from the European system of Guilds.
At any rate, the Vedic civilization was based on the Vedas, which spoke the worship of Vishnu (hence Vaishnavism) and his ten avatars (the tenth is assumed to be incarnated yet in the future). The Upanishads expanded upon the Vedas, and other great poems, such as the Mahabharata (of which the popular Bhagavad Gita is a part), further resulted in the religion's growth among the people.
The ancient Tamil documents spoke of Shiva as the Creator (with his Shakti, which can either be seen as his creative energy, his feminine side, or even his wife). The Tamil-speaking (and related languages) people were in the southeast of India, now the states of Tamil Nadu, Andhra Pradesh, and the country of Sri Lanka.
Apparently at some point, the two cultures traveled widely enough to meet one another. In what is perhaps the only time in the history of mankind, these two cultures examined each others' religions, and rather than declaring war, declared them co-equal (which through the centuries has confused even Hindu scholars into thinking it truly is a single religion).
In the background, for whatever reason the women were mostly left out of the observances of this religion, and from this rose the worship of Devi, or Shakti, which today is called Shaktism. All three groups today include male and female worshippers, but only the Shaktins have any females in the priesthood.
The fourth, and smallest, sect which makes up "Hinduism" is called Smarta or Smartism. The Smartas believe in the Vedas and other Vaishnava writings, but rather than believe in Vishnu as the Supreme Deity, they feel it is up to the believer to choose his or her primary deity from among the gods. The main effect this has had upon Hinduism has been the naming of the ultimate deity as Brahman, as he is often referred to in the Vedas, and allowing Vishnu, Shiva, and and Brahma to be seen as a tripartate form of Brahman; they can be trivialized to "preserver, destroyer, and creator", as they often are when seen by the West, or considered each and all to be full Brahman.
Around the 6th Century b.c.e., the 24th Tirthankar, Mahavira, solidified the Tirthankar teachings into Jainism, still a powerful sect despite its small size and belief non-procreation, as well as in the holiness of the tiniest creature on the planet. This is perhaps the gentlest religion on the planet, as it reveres all life and seeks to harm nothing in any way.
The next "reform movement" in Hinduism was begun by Gautama, called "the Buddha". At one point in time, Buddhism encompassed not only most of India but also much of eastern Asia. The earliest origins are clouded in history, but the Second Council (which became schismatic) was thought to have been held around 100 b.c.e. As the Buddha mostly taught the same basic spirituality but was essentially non-theistic (practice was emphasized over belief), it engulfed many other native religions in the region, but this was ultimately responded to.
In Andhra Pradesh, Tamil Nadu, and Kashmir, from 800 to 1100 c.e., new versions of Shaivite thought emerged, adopting the Vedas but promoting Shiva as the Supreme Deity and expanding upon both ancient teachings and the more modern Buddhist teachings, showing the "fallacies" in either or both, resulting in Virashaiva and Kashmir (Trika) Shaivism. In the predominantly Vaishnavite areas of Punjab and Bengal, Krishna emerged from the Mahabharata as the Supreme Form of Deity (Vishnu).
Around this time also came the Islamic invasion, which nearly wiped out Vedic teachings and destroyed thousands of books (some of which survived by having been transported out of India by Buddhist monks over centuries). Remember, by this time even the Shaivas had adopted the Vedas.
About 1600 c.e., the great teacher Arjan Dev, in Punjab, collected the greatest surviving teachings in the Hindu world, in poetry form, forming what became the Adi Granth, and created the Sikh religion using this book as their center. This book by itself preserved much of the sacred poetry from the torch of the Moslems, as the Sikhs became known as ferocious fighters and, slowly, beat back the invading Moslems. The Adi Granth was expanded by later Sikh Gurus, until, upon the death of the 10th Guru, the book itself was proclaimed the True Guru (Guru Granth Sahib).
The various forms of Hinduism have changed the West in many small ways, but are still largely misunderstood in the West. For instance, Buddhist monks created "malas", necklaces of meditation beads, well before Christ; when this came into contact with the Catholic Church, I cannot say, but it was modified into the Rosary. Madame Blavatsky in the 19th Century c.e. tried to adopt and even alter Hindusim and present it to the West as Theosophy; she hoped to raise J. Krishnamurti to be the Avatar for the New Age, but the man himself, upon reaching majority, declined the honor.
Much more could be said; indeed, much of what I just presented is in doubt and the sources may be fuzzy. Hinduism today consists of (in the view of Hindus) Vaishnavism, Shaivism, Smartism, and Shaktism, but it is easily seen how one could include Jainism, Buddhism, and Sikhism, as well as other sub-sects (such as ISKCON, a branch of Vaishnavism), into the mix.
In the last 20 years, Western Pagans have found many similarities between their beliefs and those of the various Hindu religions. The result is IndoPaganism, which was reported on in PanGaea Magazine, Spring 2007, by Devi Spring.
There is no end to this subject. It is my fervent hope that this document can be improved, especially with the addition of appropriate footnotes.
Gerald L. "Moss" Bliss, D.D.