Brotherhood of Steel
The Brotherhood of Steel is one of the first factions introduced in Fallout, and it's an important one thanks to Maximus' allegiance. It's also one of the most recognizable factions thanks to the iconic Power Armor. In season 1, Maximus becomes a squire for Knight Titus, but Titus dies in a mutant bear attack. Maximus takes Titus' armor and continues the mission, which leads to encounters with Lucy and The Ghoul.
The Brotherhood of Steel claims to restore order to the Wasteland, but their real goal is technology. They want to maintain, preserve, and regulate advanced technology. In Fallout season 2, the Brotherhood will likely continue fighting the NCR and seeking revenge on Maximus for betraying them. However, one theory suggests that the Brotherhood is connected to Caesar's Legion, a ruthless group from Fallout: New Vegas, which is a possible setting for season 2. The Brotherhood's members having Latin names adds to the possibility that they conquered the Legion and took on their beliefs.
New California Republic
The New California Republic (NCR) is another major faction in Fallout, led in part by Lee Moldaver. After the nuclear war, the denizens of Vault 15 founded Shady Sands, which grew into the NCR. Moldaver is revealed to be the leader of the NCR in 2296, and she plans to use the cold fusion capsule to restore power to Los Angeles. Although Moldaver seems to die in season 1, one theory suggests she is a clone of Miss Williams or a Synth, meaning she might return in season 2.
The NCR will likely return with a vengeance in season 2. A cameo appearance in Fallout could explain the Moldaver mystery, which could be key to season 2's story.
The Enclave & The Govermint
The Enclave and the Govermint have played minor roles in Fallout so far. In the games, the Enclave is a continuation of a pre-war American deep state organization. The show's Enclave is still mysterious, but it's clear that they are performing experiments. Dr. Siggi Wilzig, an ex-Enclave scientist, escapes with the cold fusion capsule implanted in his head.
The Govermint is a group of thugs led by Sorrel Booker. They think they have authority over the Wasteland and are connected to the Super Duper Mart's organ harvesting business. Although the Enclave and the Govermint were teased in season 1, they could be a major force in season 2.
Vault-Tec & Its Fellow Corporations
Vault-Tec is hinted to have dropped the first bombs, although some theories suggest otherwise. A meeting between Vault-Tec leaders and other corporate figureheads, including Mr. House, suggests that corporations are still vying for control after the apocalypse. With House's connection to New Vegas, the reveal about Hank MacLean, and Cooper Howard's family likely being in a Vault, Vault-Tec and its corporate allies will likely return in season 2.
Fallout season 1 is streaming on Amazon Prime Video.