This is the state of affairs before the story even begins; the opening scene shows the priestess of Apollo discovering what has happened: Orestes and the Furies are asleep in Delphi. This opening situation essentially carries over from the end of Libation Bearers, the previous play in the Oresteia trilogy. It shows us that things have yet to be resolved between Orestes and the gods over the killing of his mother Clytemnestra. Dum dum dummm.
Not only have things yet to be resolved, but also Orestes and the Furies are still going head to head. Apollo sends Orestes to Athens; the ghost of Clytemnestra sends the Furies after him. It's basically a manhunt that transcends both the afterlife (don't mess with angry ghosts) and the border between the mortal world and the gods' world (if you make sacrifices to Apollo, he'll have your back.)
Just when it looks like things are bad for Orestes—the Furies have caught up to him and caught him, literally, with their binding-song, Athena shows up and twists the plot up even more by getting the parties to agree to a trial. It looks like things aren't over for Orestes yet.
The whole problem of the play comes to a head when the Furies get Orestes and Apollo to admit that Orestes killed Clytemnestra on Apollo's command—but then Apollo argues that the murder was justified, and that Clytemnestra isn't even related to Orestes by blood, because no mother is related to her child by blood. Even if this weren't the dramatic climax of the play (which it is), it would certainly be the climax of its outrageousness.
The votes are being counted. Will Orestes get acquitted? Or will the Furies tear him limb from limb…?
The vote is even; Athena breaks the tie in favor of Orestes. From Orestes's point of view, the play is now over, and he's off the hook. He heads home to Argos, swearing that his city and Athens will be allies forever. Everything seems to be coming up roses.
Even after Orestes gets acquitted, there's still the problem of what to do with the Furies, who are spoiling for a fight, and threatening to destroy the city of Athens in revenge. When Athena convinces the Furies to become local divinities in Athens, in charge of helping the good as well as hurting the wicked, this is the final signal that everything is going to be OK. The name-change of the Furies to the "Kindly Ones" or "Eumenides" (in Greek) doesn't just bring a positive end to this play, but also to the Oresteia trilogy as a whole.