A preeminent scientist, Doctor Braun headed up many astonishing experiments in his long and varied career, but of most interest were his theories on advanced life preservation. It is said that the U.S. Army had contracted Braun to develop various methods of sustaining human life in case of an atomic war. When the war did come, Braun escaped to the safety of Vault 112 and is there to this day, his wizened and wrinkled form given new (virtual) life as a little girl named Betty.
Born in the town of Kronach, Bavaria, Germany, Dr. Braun was the Director for Vault-Tec's Societal Preservation Program, and notes from other vaults suggest that he was an outside point of contact for Overseers during the Vault Experiment, either as administrative authority or expert scientific adviser. Though early correspondence showed him taking a larger role in activation and assignment of the vaults, much of his responsibility has lapsed as he has become more immersed in, and obsessed with, the sole administration of Vault 112's virtual reality experiment.
For the past two centuries, Braun has been repeatedly creating and exploring new simulated worlds such as "Toucan Lagoon" and "Slalom Chalet," but often becomes bored and began manipulating the simulations to impose cruelty on the other vault dwellers in the simulation, as an attempt to entertain himself. A failsafe terminal in Tranquility Lane shows that Braun commissioned the "Chinese Invasion Failsafe," a means of permanently killing the inhabitants of Vault 112, but (much to his disappointment) it will not kill him, which would leave him trapped alone in the simulation.
In the current simulation world of pre-War "Tranquility Lane", Braun has taken up the identity of a little girl named "Betty" in another attempt to entertain himself - in the simulator he can be anything he wishes to be, and he happens to feel like trying out the form of a young girl. As "Betty", Braun speaks in the perfectly simulated voice of a real little girl but once he is revealed, he tends to alternate between the "Betty" voice and his real voice. Most of the residents of Tranquility Lane are unaware of Betty's nature, though a few are vaguely aware she's important and one child says "she's mean." The only person who is aware of Betty's true nature is Old Lady Dithers.
Portrayed as a narcissist and a psychopath, Braun is arrogant and has a tremendous ego, an inflated sense of his own intelligence, and anger problems. He lashes out over perceived slights, believes that he is always right, and answers even simple questions with lengthy digressions.
He also has a sadistic side, lacks any empathy for his victims in the simulation, and does not have any mercy as he views the helpless Vault 112 inhabitants as mere playthings for his own amusement. Braun views death in the simulation as a minor deterrence as he can normally resurrect them at any time he desires.
Tranquility Lane: Braun (aka Betty) is the man pulling the strings when the Lone Wanderer enters the Tranquility Lane lounger. He tasks the Wanderer with all sorts of malicious objectives, the pursuit of which nets a lot of bad Karma. Activating the failsafe gives good Karma for ending the torment of the residents in Vault 112. Doing every task of Betty's except the "Kill Everyone" task and activating the failsafe instead will result in neutral Karma.
If the Lone Wanderer tries to harm Betty in the simulation, even after completing the quest, Betty will say they cannot do that here [in the simulation], and make them pay by instantly killing the Wanderer in what appears to be a pulse blast.
As with all "real" Vault 112 residents (i.e. you, Dad and Braun - most pods, including the inhabitants, are just static scenery) in sitting state, Braun is set to both perma-unconscious (closed eyes) and ghost (you can target him, but he receives no damage from any source) state.  Because of that, even though he isn't essential, he's unkillable at all without hacking the game; to actually be able kill him, you'd have to first open his pod by PlayGroup Forward 1 on it (b724a), and then set ghost 0 on him (bd694). Even then, due to him being unconscious, he won't be targetable (though he can be damaged normally).
If the perma-unconsciousness is removed (e.g. by use of console or scripts), he only has the standard dialogue options and voice of a male NPC.
Being born long before the Great War, Braun is one of the oldest human characters in Fallout (as the captives of Mothership Zeta are older solely due to cryogenics).
Even if the Wanderer decides to run the fail-safe and permanently kill the other residents of the Vault, Braun notes that Vault-Tec failsafes prevent him from dying, perhaps forever.
Stanislaus Braun is the only person in the Tranquility Loungers that appears to be sleeping, as all the others are permanently awake.
Pinkerton stole a memory chip from Braun's terminal, claiming that it was like "stealing from a little girl".
In an interview with 1UP, Emil Pagliarulo revealed that Betty was partially inspired by the Twilight Zone episode "It's a Good Life" in which a small, willful child who was born with godlike powers is able to isolate the entire small Ohio town from the rest of the world and keep the population as his own personal playthings, tormenting, torturing, and killing them at will.
The following is based on unverified behind the scenes information and has not been confirmed by canon sources.
Betty also has similarities with AM, the military supercomputer from Harlan Ellison's 1967 short story I Have No Mouth, and I Must Scream. AM, after becoming self-aware and almost wiping out humanity, keeps only a handful of humans alive for the sake of torturing them.
Betty further shares similarities to a chapter of the 1960's Sci Fi Novel by Philp K. Dick "The Three Stigmata of Palmer Eldritch". While inhabiting in a drug (Chew-Z) induced dream world that links multiple minds, Palmer Eldritch takes the form of a little girl named Monica to hide his identity from and manipulate character Leo Bulero, his nemesis, who is also momentarily inhabiting that world with Palmer. That the little girl was secretly Palmer the whole time is revealed to Bulero and the reader simultaneously.
End of information based on unverified behind the scenes information.