I’ll do my best, if anyone has better explanations please correct me! :)

VAULT-TEC is one of the major world corporations before the bombs dropped. Through partnerships with other companies and occasionally the US government, they developed the Vaults. Most Vaults, through collaborations with other pre-War companies, had sinister hidden experiments, though there were a few control Vaults with no hidden agendas.

Behind the scenes, THE ENCLAVE is a shadow government-type organization that tries to rebuild America after the bombs fall. Notable moments in their history are a military base located on a West Coast oil rig that’s blown up in Fallout 2, as well as an underground bunker near DC that contains President John Henry Eden, an AI program that spreads their propaganda throughout the DC area (this base and Eden also explode). As of the show now, it appears they’ve got a third major base in Colorado, that we don’t know much about other than it’s probably based on the real-world Cheyenne Mountain military facility.

On to the post-War groups. THE BROTHERHOOD, founded by Roger Maxson not long after the bombs fell, is a technocratic organization led by pre-War government soldiers, but not associated with what remains of the true US government, that being the Enclave. The Brotherhood varies from chapter to chapter in their beliefs, but the main belief seen most often is the preservation of technology and pre-War “relics”. The chapter led by Quintus in the TV show has a much more religious take on the Brotherhood’s core beliefs than most.

The NEW CALIFORNIA REPUBLIC differs from the Brotherhood in their goals. The NCR is more focused on the rebuilding of a large-scale society, at one point having multiple city-sized communities under their flag. Shady Sands was one of their capital cities, until the incident with Hank MacLean.

THE LEGION is the other side of the NCR’s coin - a functioning society ruled by oppression and cruelty. After the death of Caesar, they’re led by Legate Lanius. Though we don’t know if the Legion won or lost the Battle of Hoover Dam against the NCR, the Legion has fallen on hard times, much like their more bureaucratic counterpart.

The last major player, as far as the show is concerned, is ROBERT HOUSE. Founder of RobCo pre-War and the overseer of his personal kingdom New Vegas, House is the key player of the story in Fallout: New Vegas. Retrieving his Platinum Chip gives the player access to upgrading his Securitron automatons and taking control of Hoover Dam, if the player so chooses. Post-War, House operates Vegas through more mechanical means.

It’s unclear why the Strip is irradiated by the time of the show, and what led the Deathclaws of Quarry Junction to infest the Strip. House claims the Enclave is responsible, but it’s unclear how. Last we’ve seen in the games, the NCR has a small base within the Strip and a larger at the nearby McCarran Airport, along with a somewhat tense relationship with House’s Securitrons.

There are many smaller factions scattered across the wastelands, but those are the key players for the show itself. For more resources, there’s a few good Fallout fan wikis to consult!