The Orestia trilogy (more or less) begins with King Agamemnon returning from the Trojan War. Queen Clytemnestra has taken up with a lover from a rival clan. She’s also pissed that her husband sacrificed their daughter, Iphigenia, to win favorable departure winds to sail to the war ten years prior. Queen kills King.
Daughter, Elektra, points out to brother, Orestes, that a son is socially and morally obligated to take revenge for his father’s murder. They give libations to the gods and to the Earth and he kills mom. Since nothing is ever straightforward, there’s nothing worse in Nature’s eyes than killing your own mother, the Harpies (aka, The Eumenides) rise from the ground and pursue Orestes. They want BLOOD!
Orestes arrives at the Temple of Apollo with the Harpies hot on his heels. He heard that he can ritually wash off the pollution of his deeds there. Unfortunately, the pig blood doesn’t prove a close enough cleansing agent for mom blood to assuage the Harpies. Apollo, having failed, turns to his sister, Athena.
She appeals to The Eumenides and convinces them to put the matter to democratic vote, with Athena playing Vice President in The Senate in case of a tie. Question posed: Was Orestes justified in killing Mom to revenge Dad, or is Mom-killing a crime against Nature greater than Dad-killing? The Demos vote ties, and Athena favors Orestes.
The Harpies are enraged. They want their blood. Athena strikes a deal: A law shall be passed that prevents any Greek from killing another Greek—and the Harpies shall be the weight and force behind that law. The Eumenides are then given new vestments—literally “invested” with a new power—signifying their new role in civil society. It is then that reciprocal violence is ended, vengeance is assumed under The State, and Athens is founded.
Note that this story takes place after The Iliad and The Odyssey—after all the stories of righteous vengeance. Aeschylus wanted the Athenians to think of themselves, to imagine themselves outside of or playing a role transcending vengeful violence and establishing civil society.
A shitty Cliff Notes version, but the idea is that, for civil society to exist at all, vengeance cannot be, must not be allowed to exist in the hands of the individual least violence spiral out of control. Rule by Law is what allows civilization to exist. And it is only the State that shall dole out vengeance.