Sargon's species

Sargon's species was a humanoid race. They explored the galaxy and colonized several planets, possibly including Vulcan, about 600,000 years before the 23rd century. About 100,000 years later, they achieved such great mental power that they began to regard themselves as gods. War broke out, "a struggle... for such goals, and the unleashing of such power, that you could not comprehend," as Sargon describes the event. The war tore the atmosphere from their planet, ultimately extinguishing all life save for a relative handful of representatives. Again according to Sargon, "only the best minds were chosen to survive." It is not clear by what criteria Sargon's people made this judgment. Those chosen transfered their consciousness into receptacles; globes approximately 50 cm in diameter. Sargon's' receptacle occupied a special pedestal in the main chamber of a cave constructed 112.37 miles beneath the surface of the planet. From here, he carried out the task of probing the heavens for starships representing a race advanced enough to save his people from oblivion. The remaining dozen or so of his peoples' receptacles occupied niches in a smaller side chamber.

By 2268, only Sargon, his wife Thalassa, and Henoch (a member of the opposing faction) still survived. In that year, the USS Enterprise (NCC-1701) discovered the spheres containing the aliens' consciousnesses. 

Powers and abilities
The aliens displayed various abilities:
 * Sargon affected the Enterprise instrumentation sufficiently to lure the ship to his world from a great distance.
 * At closer range, he could speak telepathically, reading the thoughts of those in orbit and replying in kind. Neither Henoch nor Thalassa demonstrated this power.
 * Sargon could, at closer range, neutralize all ship's power (without damage), and enhance the ship's transporters and communications systems to permit transfer of matter and information between the ship and the landing party located more than 112 miles below the surface of his planet.
 * Sargon persuaded Ann Mulhall that she had received an order to report for landing party duty. It was only when Kirk asked her from whom she'd received the order that she became puzzled, but even then she remained convince of the order's legitimacy.
 * All of the aliens could transfer their minds to suitable host bodies, and back again. Sargon could briefly house his mind within the machinery of the Enterprise.
 * All of the aliens could direct energy at other beings and objects. Thalassa used this power to torture McCoy for a few seconds before realizing her moral error. Sargon used it to destroy the receptacles that once housed their minds.
 * All of them demonstrated great skill in engineering; Henoch's understanding also encompassed pharmacology. It is likely that all of them possessed superior intellect relative to Humans.
 * Sargon, or possibly Sargon and Thalassa acting together, placed two consciousnesses within a single body.
 * Sargon could hide his presence from others of his kind; he successfully prevented both Thalassa and Henoch from realizing he still lived until he chose to reveal himself. Thalassa did not understand this until Sargon explained that "he had power Henoch does not suspect."
 * Henoch could telepathically torture Humans in some invisible fashion. He used this power to "make an example" of Uhura, presumably when she refused to follow his orders.
 * Henoch could make multiple Human individuals experience sourceless but intense pain.
 * Thalassa also referred to powers "Sargon did not permit them to use" because he believed them too tempting. It is possible that Henoch's unpleasant offensive tactics were among these.

One important limitation seemed to be the requirement for a suitable vessel; these aliens could not exist without either a body or a receptacle of some type. This differentiates them from true noncorporeal life such as the Organians or the Q. In addition, their presence placed great strain upon Human and Vulcan host bodies, requiring frequent injections of a "metabolic reduction agent" to compensate. In time (months or years), host bodies grew accustomed to the aliens' powerful minds, however.

Individuals

 * Henoch
 * Sargon
 * Thalassa

Apocrypha
Greg Cox's novel Q-Strike from the Star Trek: The Next Generation - Q Continuum series suggests that the creature known as (*), aka the Beta XII-A Entity, was the cause of the war among Sargon's people which destroyed the planet's atmosphere.