By the time the bombs fell, the Vault-Tec corporation had basically become an extension of the U.S. government. They leveraged this judicial leeway, along with all the evil of any San Francisco tech startup, into massively unethical psychological experiments, like pumping people full of hallucinogens. Not the fun ones, either, but the kind that turn you into mindless killing machines.
You can imagine, then, that the 122 fallout shelters -- Vaults -- that the company scattered across the country weren't for preserving wholesome suburbanites. They were, in fact, part of the "Societal Preservation Program." This wasn't meant to protect Vault residents, but as an excuse to gather more unwitting "volunteers."
Government officials, and certain Vault-Tec employees intended to use this data for what they correctly predicted would be the end of life as we know it. Unfortunately for this shadowy conglomerate, however, Judgment Day came sooner than they predicted. A society that still listens to The Ink Spots in 2077 can only sustain itself for so long, after all.
Speaking of doomsday preppers with too much money, The Enclave is what remains of the group Vault-Tec was meant to save. Technically, The Enclave is a rump state -- an useless, abandoned government that still tries to exert influence over citizens that have long-since moved on. In this way at least they're very similar to the real U.S. government, and style themselves accordingly.
By necessity, members of The Enclave are descended from former members of the U.S. government who had real fallout shelters built in secret. When they're not acting like a conspiracy theorist's wet dream, The Enclave are up to more direct wrong-doings. They see the irradiated and mutated inhabitants of the Wasteland -- i.e. everybody -- as "impure." That puts them at odds with less technologically advanced factions (i.e. everybody), which they oppress and exterminate accordingly.
Depending on who you ask the Brotherhood of Steel is either a bastion of post-apocalypse society, or its most disappointing failure. The Brotherhood (which does include women) has more than enough pre-war technology to aid the unfortunate skells of the Wasteland. Just not the inclination.
After witnessing Vault-Tec-like human experimentation on behalf of the U.S. government, several members of its military went rogue. This was before World War III filled the world with sentient fungus and homicidal can openers, but only just. The ensuing war left these rogue soldiers without a mission, but a whole lot of grade-A military murder machines. Rather than use or share this technology, they began hording it in the hopes of one day restarting civilization.
You could give the Brotherhood points for not turning their signature power armor and miniguns on weaker individuals, sure. But that would ignore the many, many times they turned a blind eye to those same unfortunates as they were gunned, gulped, and gutted by the world's many new dangers. Hundreds of years after the Brotherhood was founded in protest of human rights violations, the organization now cares more about machines than people. But at least their armor makes for really neat box art.