Greek god
The Greek gods were an immortal alien species, native to Pollux IV in the Beta Geminorum system, who possessed powerful psychokinetic abilities.

History 
Self described as "a gallant band of travelers," this species visited the Mediterranean region of ancient Earth around 2700 BC. Their psychokinetic abilities made these beings appear god-like to the simple shepherds and tribesmen of early Greece, and so these beings were worshiped as the Greek gods as they resided on Mount Olympus, forming the basis for Greek mythology. ()

Songs of praise to the Greek gods and accounts of their deeds were included in the Homeric Hymns. ()

Humanity gradually outgrew the need of these gods until they were only memories. Unable to survive as a memory, these space travelers departed Earth and returned home to Pollux IV. There, they found it an empty place without worshipers, but they lacked the strength to leave. So they waited. Eventually, all of the gods save Apollo lost hope, and departed their corporeal existence. They "spread themselves upon the wings of the wind… thinner, and thinner, until only the wind remained."  

In 2267, the  encountered the species on Pollux IV, where Apollo remained. During this encounter, Apollo attempted to force the crew of the Enterprise to worship them once again, and took Carolyn Palamas as his queen, promising to make her a goddess.

This, unfortunately for Apollo, was a state of affairs unacceptable to men. Captain James T. Kirk and his landing party schemed continuously; between them, and members of the crew still in orbit, they were eventually able to overcome Apollo, and destroy the source of his power. With his temple gone and his power broken, Apollo spread himself upon the wind, imploring his fellow gods to take him, and vanished. ()

The memory of the Greek gods, namely , lived on with the Platonians, another alien species that visited Earth during the tail end of the Greek civilization. This was discovered, again, by the crew of the USS Enterprise, during their visit to Platonius in late 2268. ()

Physiology 
 
As represented by Doctor Leonard McCoy's tricorder scan of Apollo's corporeal form, who for all intents and purposes appeared Human, registered as a normal humanoid, with just a few variations. Most notable was that he possessed an extra organ in his . This organ allowed him to generate and control energy by channeling it through his body and direct it by acts of will. This organ allowed the Greek god to tap a powerful field of energy, generated within his temple. However, scans from Enterprise were unable to identify his presence.

Though possessing great power, these Greek gods were weakened after having overextended their powers, requiring a brief period of rest during which their vitality was restored. As their form of sustenance, these gods thrived on love, admiration, worship, in much the same way Humans needed food to survive. After time without this needed sustenance they gave up their corporeal existance. Though immortal, these gods did not die, they simply reached "a point of no return," and "returned to the cosmos on the wings of the wind."

As a corporeal, Apollo's abilities included that of altering his physical own form as well as the forms of material objects in his vicinity. 

He could hurl lightning-like energy bolts, interfere with Federation technology (i.e. disable the ship's "transportation device", and hand phasers, as well as jam communication frequencies), transform s into ancient Greek clothing by simply waving his hand, transport himself and others from place to place, and alter the local weather patterns. 

Though never fully demonstrated, Apollo claimed he could wipe out a man with a gesture, and bring him back again just as easily. 

The Greek god's senses extended great distances; while on the surface of his world, Apollo was able to sense happenings on the bridge of the Enterprise, and project his image and voice into her equipment for the purpose of conversation.

Perhaps his most tasking feat was the generation of a powerful field of energy that appeared as an enormous, translucent, green hand, capable of holding a  in place, and resisted most attempts to break and disrupt it. Though conventional in composition, it operated on unusual wavelengths, generated by Apollo. This field was powerful enough to stop the ship, and impulse engines could not break the ship free.

In addition to merely holding the ship, Apollo could exert pressure beyond the ship's tolerance, over 1,000 GSC of external pressure, threatening the structural integrity of the ship. In addition to his demonstration, he explained that once everyone was off the ship he would crush it's empty hull.

Attempts to push it away with the tractor beam, and to reverse its polarity were equally unsuccessful in disrupting it. The force field was eventually pierced with powerful bursts of M-rays on carefully calculated wavelengths. Although this did not destroy the field, it enabled the Enterprise to fire her phasers through it.

After the temple was destroyed, Apollo demonstrated that he still possessed some intrinsic abilities when he expanded his corporeal form. However, from his reaction to the temple's destruction, he expended the last of his stored energy, before he himself gave up his corporeal existence. ()

Greek gods
Aphrodite, goddess of love, beauty, and sexual desire
Apollo, god of music and medicine
Artemis, goddess of hunting and Apollo's beloved sister
Asclepius, god of medicine and healing
Athena, goddess of wisdom
Atlas, a 
Hephaestus, god of fire and the forge
Hera, queen of the gods 
Hermes, messenger of the gods
, god of the shepherds
 Persephone, daughter of Zeus
Zeus, king of the gods; father of Apollo and Artemis

Appendices

Background information 
The  dated the visitation of the Greek gods to around the year 3000 BC.

It is never revealed how long Apollo and the other Greek gods remained on Earth, with Apollo stating only that "it has been five thousand years".  The earliest references to Greek mythology historically date from around 900BC, which would have been well after when Apollo states the Greek gods had visited Earth.  Dialogue from Captain Kirk speculates that Apollo and his companions were the "basis" for Greek mythology, implying that there was a difference between the time the Greek gods actually lived on Earth and the eventual myths and legends which developed about them over a thousand years later.

StarTrek.com has theorized that the Olympian gods may have also visited  and Romulus, centuries earlier before coming to Earth, to explain how words like Vulcan, Romulus, Remus, Centurion and Senate, etc. came to be used by all three races. StarTrek.com also went on to speculate that the Gods may have been responsible for separating the Vulcan and Romulan races.

Apocrypha 
In the comic book What Fools These Mortals Be, the crew of the USS Enterprise encounter a planetoid resembling ancient Greece that is ruled by the Greek gods. The gods subject the landing party to various events from The Odyssey, including encounters with the cyclops, sirens, Aeolus, Scylla and Charybdis. The landing party manage to embarrass Zeus causing the gods to leave the planet.

In the Star Trek: New Frontier novels Being Human and Gods Above, it is revealed that the Greek gods were the same beings as the Egyptian and Norse gods, just known under different names. Despite Apollo's belief that they had all perished, they did in fact survive. The gods are revealed to be energy beings assuming a humanoid form, who feed on the worship of their followers. Apollo's energy supply in the temple had been limited, which is why he was so desperate for the Enterprise crew to worship him. The gods offered ambrosia to anyone who would worship them and attacked those who didn't, but were eventually defeated and sealed in another dimension through the combined efforts of Old Father and Mark McHenry, a descendant of Apollo.

External links 
 
 

fr:Dieux grecs
it:Dio greco
Non-corporeal species
Species