Psychic (Fallout)

A psyker is a person who posses a certain paranormal power either from FEV like the master's experiments and Melchior, Radiation like the Beastlords, natural selection like Hakunin or 200 years of seclusion in a glass jar like Professor Calvert.

Fallout
After being dipped in FEV vats, absorbing some other people and being connected to the LA Vault computer, the Master gained some psychic powers the extent of which is unknown. He was able to attack humans mentally, although his attacks could be repelled with proper training. The Master was able to somehow bring forth psychic abilities in certain humans after they were injected with FEV (usually the pineal gland, but also the Amygdalae or Medulla), but most of the experiments were failures (resulting in insanity) or used to line the corridor of revulsion. There were four psykers in the Master's lair:
 * Wiggum was electrokinetic-dominant,
 * Lucy was telekinetic-dominant and a minor photokinetic,
 * Moore was pyrokinetic-dominant,
 * Gideon was a receiving-telepathic-dominant (without the ability to control his telepathy, requiring the Psychic nullifier to block incoming thoughts) with minor photokinetic abilities.

All four of them died when the Cathedral was vaporized in nuclear fire.

Fallout 2

 * Hakunin, the Arroyo shaman, apparently also possessed some telepathic abilities, as he was able to speak to Chosen One in his dreams (The Chosen One was able to receive it and record it with his PIP Boy).
 * Melchior, miner from Redding, was able to summon dangerous animals (deathclaws, fire geckos, mole rats), when he became mutated by FEV in Mariposa Military Base.

Fallout 3

 * Professor Calvert - one of the residents of Point Lookout, he achieved psionic powers after an experimental procedure. When his brain was extracted from his body (in the same fashion as for robobrain models) and put into a jar, he gained an ability to communicate with people all around the region by entering their minds. To a person, it appeared like a mysterious voice in their head.
 * Bloomseer Poplar - is one of the members of the Treeminders and can predict your future, she describes the settings of some of the major quests you can do like The Superhuman Gambit, "I see a battle between insects and industry, and the men who control them...".

Fallout Tactics

 * A group of tribals known as the Beastlords had a kind of psychic control over animals.

Fallout: New Vegas

 * The Forecaster is a child who can see into the future, but can't control his powers. He needs his "medicine" to keep him from over-thinking, thus helping control his headaches. For 100 caps, though, he will forgo his medication and look into the future for the player.
 * Nightkins inside Black Rock cave seem to have some paranormal knowledge of events and characters, if you've completed the quest Crazy, Crazy, Crazy by repairing Rhonda they will tell the "voices" said things that will happen in the main storyline, as well as special comments about the companions, EG: If you bring Craig Boone as a companion, they will comment "silent man knows more about the spiked crowns than what he says" referring that boone knows more about great khans than what he says (the bitter springs incident).

SPECIAL
The ability to resist the Master's psychic attacks is represented in the SPECIAL system by the Mental Block perk.

Appearances
Some kind of psychic powers appears in every Fallout game, although Chris Avellone believes it creates narrative difficulties for the Fallout writers, as stated in Fallout Bible 3. Avellone says:
 * For Fallout fan-fiction purposes, you are welcome to make use of the psykers and their potential from Fallout 1, but I'd be careful - the psykers in Fallout 1 show some pretty over-the-top mutations that could take the world to Childhood's End faster than you can say "uh, his eyes are glowing?"