Psychic

A psychic is a person or creature that possesses supernatural or paranormal powers.

Background
Either through the ever-present background radiation of the world, the release of and further experiments with FEV, and in some instances of technological experimentation or ancient alien artifacts, individuals have gained powers of the mind. Many psychics have been exposed to FEV, but there are also psychics that have no known exposure to FEV, but no matter the cause the effect is nearly always random. The powers could range from the benign of clairvoyance and telepathy, to more extreme projections of power like pryo and electrokinesis.

In the cases of human psychic powers, the instances are often detrimental to the psychic's psyche causing insanity of varying degrees, which was the case for all of the Master's experiments and Lorenzo Cabot's encounter with an Ubar artifact. Naturally born psychics are more adapted to their phenomena, but cases exist where the psychic must take precautions to avoid detrimental affects or the use of it causes intense strain and possible death. Those outside the range of human (or without a body) often do not suffer any detrimental affects and are adjusted to any stress and stimuli that psychic powers bring.

Psychics in the wild are incredibly rare, however. The instances of such individuals are few and far between and skepticism for the paranormal is well and still alive.

Fallout

 * The Master gained psychic powers after absorbing previous failed experiments in producing his "psykers" and now possess a permanent area of revulsion around him that assaults the minds of all that enter.
 * The ability to resist the Master's psychic attacks is represented in the SPECIAL system by the Mental Block perk.
 * The Master also attempted to create similar psychic creatures, but most of the experiments were failures resulting in insanity. Four individuals in the Cathedral successfully gained the abilities, referred to as psykers. These characters include Wiggum, Lucy, Moore, and Gideon. A device known as a psychic nullifier was utilized to block incoming thoughts.
 * Chuck in Adytum will offer to read the Vault Dweller's tarot cards.
 * Tycho mentions that he doesn't believe in psychics or telepathic abilities.

Fallout 2

 * Hakunin, the Arroyo shaman is able to speak to the Chosen One in their dreams.
 * Melchior is a super mutant who has the ability to command creatures such as deathclaws, fire geckos, and mole rats.

Fallout 3

 * A member of the Treeminders named Bloomseer Poplar can share what she sees in the Lone Wanderer's future.
 * Professor Calvert underwent an experimental procedure to extract his brain in an attempt to extend his longevity. Because of experiments involving bio med gel composition, he has the ability to project his thoughts into the minds of others, and wishes to use this ability to achieve "psychic domination."

Fallout: New Vegas

 * The Forecaster is a child who can provide visions of the future. If he utilizes this ability too much, he suffers from headaches and wears a psychic nullifier, which he refers to as his "medicine," to avoid the side effects.

Fallout 4

 * Mama Murphy is a woman who can foresee events after taking chems, an ability she calls "the Sight." She can share these visions with the Sole Survivor.
 * Lorenzo Cabot has powers stemming from an ancient artifact that he discovered during an archaeological dig in the Rub' al Khali. He describes the abilities he has gained, such as exploring the memories of the ancients of Ubar, communing with those who founded our civilization. He also gained a number of other supernatural abilities, such as halted aging, telekinesis, and increased strength and resistances. Some of these abilities can be artificially transferred to others, albeit temporarily, through the consumption of special serum derived from Lorenzo's blood. This has been the case for his family and, may potentially be so for the Sole Survivor.

Fallout 76

 * The Flatwoods monster is able to warp the mind "of all but the most iron-willed."
 * The Interloper sent visions to Jeff Lane, beckoning him to Lucky Hole Mine in order to become its conduit.

Fallout Tactics

 * The Beastlords have the ability to command creatures.

Behind the scenes
In regards to paranormal/supernatural content such as psychics, Joshua Sawyer stated via Tumblr that he thinks, "a li'l bit is okay and it feels like if we're going to accept some of the wacky science of the setting, we're already off the rails of strict realism. That said, I think it's best if it feels alien and remains more or less unexplained."