A psyker is a psionic individual in the Warhammer 40,000 universe, drawing energy from the Immaterium and manipulating it to produce supernatural effects. Certain species have a greater number of psykers with greater psionic powers. The Tau and Necrons lack any psionic abilities at all.
A certain percentage of Humans in the Warhammer 40,000 universe are born with psionic abilities. Generally these powers are quite low level and would otherwise be unnoticed, but in a few individuals these psionic powers can be formidable, and any unwary psyker attracts the attention of daemonic warp entities. Consequently, psykers are generally feared and mistrusted in the Imperium as they are often associated with cults, heretics, and unfortunate supernatural incidents.
By far the most powerful psyker in existence is the Emperor, who is practically a god.
In the Imperium, the Adeptus Astra Telepathica has responsibility for regulating psykers. Turning psykers over to the Imperium is a major responsibility of planetary governments. The psykers are transported aboard craft known as Black Ships to Holy Terra to either be consumed by the Emperor, help power the Astronomicon, or to undergo soul-binding rituals that will allow them to serve as Astro-telepaths or as carefully monitored battlefield specialists. Most chapters of the Adeptus Astartes employ specially trained psykers known as Librarians.
Pariahs are humans who are essentially "anti-psykers". They have no psychic ability and are resistant or even immune to psychic phenomena. Stronger pariahs inhibit the abilities psykers in their vicinity. Psykers and even ordinary humans find the mere presence of a pariah extremely disturbing.
The Eldar are a race with well-developed psychic abilities. While all Eldar have the ability to manipulate energy from the Immaterium for supernatural effects, the circumstances of the fall of their civilization make them unable to use their psychic abilities freely. Any attempt to directly tap the energy of the Warp would result in their immediate consumption by the Chaos god Slaanesh. To avoid this fate, Eldar Seers and Warlocks use psychic foci called "runes" to moderate their access to Warp energy, allowing them to perform specifically prepared psionic feats without being destroyed or possessed at a cost in versatility.
As psykers draw their power from the Warp, which is the domain of the Chaos Gods, psykers dedicated to the cause of Chaos tend to be exceptionally powerful, with psykers dedicated to Tzeentch topping the list. These "sorcerers" and "witches" frequently have psychic abilities associated with their patron gods: sorcerers dedicated to Slaanesh, for instance, often have the ability to manipulate the pleasure centers of other beings, while sorcerers dedicated to Nurgle often have the ability to cause virulent diseases (or their symptoms) to manifest. The powers of chaos psykers are typically dependent on how well they serve the interests of their patrons; sorcerers who displease their patron god are likely to find that their powers fail them.
The forces of Khorne generally don't have psionic powers; Khorne considers the use of such abilities cowardly. Instead, power from Khorne manifests as superior physical attributes that will aid the servant in combat.
Psionic orks are known as Weirdboyz. While they have minor innate powers, they become particularly powerful in the presence of large numbers of other orks. Weirdboyz absorb psychic energy from nearby orks, and they can release this energy for a variety of supernatural effects. Few weirdboyz have substantial control over their abilities, though, so they have "minderz" to control their exposure to other orks and help them direct their manifested powers (often by literally aiming the weirdboy's body at the enemy). Weirdboyz that absorb too much psychic energy from other orks can suffer severe consequences, up to and including head explosions.