The Brotherhood of Steel: The Brotherhood of Steel (BOS) is a techno-religious organization, with roots in the US military and government-sponsored scientific community from before the war. The BOS is mostly composed of the descendents of those military officers, soldiers, and scientists, but aside from some outlanders among their ranks, the BOS is as close to pure strain humanity (prime normals) as you're going to find outside of a Vault.
The ranks of the BOS is generally recognized as being composed of the best and the brightest... which means the BOS is a really small organization, at least compared to NCR. They make up for this with their frightening arsenal of pre-war and post-war technology: They have laser weapons, power armor, surgical enhancements, combat implants, and a squad of Brotherhood Knights have the ability to erase an entire town from a map without a scratch.
The Brotherhood are generally good guys, but they have their faults -
(1) they don't care for mutants,
(2) they worship technology, and in many cases, put it above human life,
(3) they don't like to share their choicest technological bits, despite the obvious benefits their technology could bring to the wasteland.
It’s commonly accepted within the Brotherhood that the people of the wasteland are not responsible enough to use (and maintain) all of the technology the BOS has at their disposal. They are known trade some of their technologies with frontier communities and NCR states, but they keep the more sensitive technologies to themselves.
It is believed that the current HQ of the BOS is the Lost Hills bunker southwest of the NCR capital, but at this time, the BOS is spread across the wastes in small bunkers and installations hidden from the eyes of common folk – finding them all and wiping them out would be a difficult and dangerous task.
The BOS is divided into different ranks: Initiates are trainees who are expected to perform well enough in the training process to be promoted to Squires. After proving themselves, Squires are promoted to Knights. After many years of service and experience, the best Knights are promoted to Paladins - the pinnacle of the Brotherhood military. Paladins who survive to their later years become Elders, and they number among the Brotherhood ruling council.
It is also possible to serve the Brotherhood as a Scribe. Scribes are responsible for copying the ancient technologies, maintaining the current technology and even experimenting with new weapons and other useful devices. Scribes rarely leave the safety of the BOS bunkers, but they are sometimes called into the field to examine a piece of technology or perform a task beyond the skills of the Brotherhood soldiers.
It is said that the BOS symbol, broken down, represents each of these orders. The sword represents the Paladins, the wings represent the Elders (the “wings” control the movement of the sword), the large gear represents the Knights, and the two smaller gears represent the Scribes and the Squires, whose services keep the Knights supplied with the information and the manpower to get their jobs done. No one is sure what the circle means, however.
Credit to Chris Avellone and the Fallout Bible