Most of what is currently known about the DHARMA Initiative comes from an orientation film, found inside an underground bunker on the island, that outlines the purpose and instructions for Station 3: The Swan. The film has a copyright date of 1980 and is designated as the third of six such films.
It states that the Initiative was founded in 1970 by University of Michigan doctoral candidates Karen and Gerald DeGroot and financed by the Hanso Foundation, apparently composed of a group of "scientists and free thinkers" from around the world who were brought together at a "large-scale communal research compound" on the island to conduct research in various disciplines, including meteorology, psychology, parapsychology, zoology, electromagnetism, and a sixth discipline that the film begins to identify as "utopian social . . . ." before being cut off.
The DHARMA Initiative has placed several research stations around the island, which take the form of hidden, underground facilities or bunkers. The first to be discovered by the survivors is "Station 3" or "The Swan" which they refer to informally as "the hatch" and have since occupied. Three additional stations have since been visited: "The Pearl," "The Arrow," and "The Staff."
Founded in the early 1970's as an attempt to "save the world as we know it", the Initiative was founded by Gerald and Karen DeGroot, two doctoral graduates from the University of Michigan, and funded by Alvar Hanso. Their goal was to solve the fundamental principals of the "Valenzetti Equation", an equation that could apparently predict the end of the world, through various apocalyptic events (nuclear warfare, overpopulation, etc). The core values of this equation are 4, 8, 15, 16, 23, 42, and the DI (Dharma Initiative) scientists were working on changing these values so that, apparently, "the one true way" could be found.
The Dharma Initiative worked on a mysterious island located somewhere in the Pacific, though it is now known to be able to move through time/space. The DI found the island by means of their "Lamp-Post" station located in Los Angeles.
Notable heads of the Initiative include Horace Goodspeed, a hippy-like character, and Pierre Chang who features in the several orientation films. Dr Chang assumes several aliases, such as Mark Wickmund and Marvin Candle, when filming these instructional videos, but it is unclear why.
Using several bunkers and research compounds, the DI performed many experiments in electromagnetism, zoology, medicine, etc (even Time itself, deep beneath the island in the Orchid Station). It is unknown at the moment if any of the mysterious phenomena seen on the island (whispers, smoke monster, hallucinations, etc) are a product of the DI's experiments, or have always been a part of the island, though it is known that the polar bears were being experimented on in the Hydra Station.
The Initiative maintained contact with the real world via the Flame Station, and could leave at will by submarine.
The DI had an uneasy truce with the indigenous people of the island, "The Hostiles" as they were known, though occasional skirmishes broke out. However, the DI were eventually wiped out by the "Hostiles" following the "purge", and they subsequently took over operation of most of the stations and living quarters.
The Dharma Initiative are still pretty much a mystery, but Season 5 of Lost has filled in much of their back story.
Goodspeed: Sorry, but you're not Dharma material.