What are the belief systems in ancient Greece? Please explain the belief in spirit possession on Athena's divine internetion.

In answer to your question with regard to the belief
systems in ancient Greece, the people of the time believed that the gods (Greek gods,
not Roman gods) were in charge of the destiny of all
men.

The people believed that the Titans had been defeated
by Zeus. They believed that the gods often came down from Mount Olympus and played
"tricks" on unsuspecting humans. Heroes from the myths were often the result of a
liaison between a god and a human, that tended to result in the birth of a demi-god like
Heracles. (Roman mythology's version is "Hercules.")

The
people explained naturally occurring processes by looking to the gods. For example,
Persephone, was the unhappy queen of the Underworld. She was required to spend half of
her year in the land of the dead. When winter came, the Greeks believed that this was
when Persephone returned to the Underworld.  When spring arrived, they believed that
Persephone had returned from the Underworld and would remain until the change of seasons
again.

Because the people of ancient Greece believed that
the gods could reward or punish them at will, they often would call upon a certain god
for protection, and/or would worship in a temple that was dedicated to a single deity.
For example, the people built a temple to honor Athena.

The
gods caused crops to fail, for thunder to sound and lightning to flash; they punished
the wicked, vain and foolish. The ancient Greeks believed monsters like the Medusa and
Polyphemus—the Cyclopes, both results of some kind of involvement or intervention by the
gods.

Greek mythology is filled with stories of gods who
often had the same characteristics as humans, and the ancient Greeks fashioned a great
deal of the things they did and feared on the stories of the Greek
gods.