The play begins outside the temple of Apollo in Delphi. This temple was home of none other than the famous Oracle of Delphi, where the god was thought to dispense wisdom through the mouth of the priestess of the temple. In the opening scene of the play, this priestess is just showing up for work.
When she opens the temple doors and goes inside, however, she finds something terrifying: a man armed with a sword is sleeping inside; surrounding him, also sleeping, are the Furies, horrible goddesses of vengeance. (What the priestess doesn't know, but the audience does, is that the sleeping man is Orestes, who has come to Delphi to be purified after killing his mother, Clytemnestra, at the end of the previous play, Libation Bearers.) The priestess is terrified at what she has just seen, and runs away.
At this point, two figures emerge from the temple. One is Orestes, who has just woken up, and the other is… the god Apollo. As if this weren't proof enough that Orestes and Apollo are best buds, Apollo now tells him, "Hey, I've got your back. Go to Athens and ask the goddess Athena for help. There you will have a trial, and I will defend you. Don't worry: everything's going to be cool." That sounds good to Orestes, and he heads offstage in the direction of Athens. Apollo heads offstage shortly afterwards.
But then, just when things started to look too good to be true, who should appear but… the Ghost of Clytemnestra, Orestes's dead mother. Clytemnestra heads inside the temple of Apollo and wakes up the Furies. She whips them into a frenzy and tells them to go get Orestes. Then the Ghost of Clytemnestra leaves, but the Furies are still furious… so much so that they start dancing and singing a song about it.
At this point, Apollo reappears, now armed with a bow and arrow. He tells the Furies to scram. But the Furies aren't about to back down without a fight. They get in a big argument with Apollo, whom they accuse of being the one who got Orestes to kill his mother in the first place. Apollo admits as much, but says that what he did was justified. After some back and forth about this, the Furies finally head offstage, headed for Athens. They are determined to hunt Orestes down and bring him to justice—i.e. tear him to shreds.
After Apollo exits, the stage is briefly empty. A moment or two later, Orestes comes back onstage; his opening words—a prayer to the goddess Athena—show that, presto-chango, the scene has switched from Delphi to Athens. He doesn't have very long to catch his breath, however, before the Furies appear onstage as well. They are tired of chasing Orestes around. Orestes barely has time to make one last, desperate prayer to Athena before the Furies sing their "binding song." What's that? Just what it sounds like: a magical song that makes Orestes stuck fast, unable to move.
Things are looking bad for Orestes, but just then the goddess Athena herself shows up, in answer to his prayers. After figuring out what all the commotion is about, Athena suggests that they put the matter to a trial. The Furies and Orestes both agree. Surprisingly, Athena says that this matter is too big even for a goddess to decide: she says that she is going to gather a jury of Athenian citizens to hear the trial; she will simply sit as judge. While Athena goes off to round up the jurymen, the Furies sing a song expressing their anxiety about this whole trial business. They would much rather tear people to pieces than piece together legal arguments.
When the Furies are done singing, Athena comes back onstage, with the jurymen and a herald. Then, as if on cue (okay, so it's a play: it is on cue), Apollo shows up, and announces that he will act as Orestes's defense attorney. Then, after Athena makes a speech about how the law-court she is now establishing will last for all time, the trial begins.
First, the Furies cross-examine Orestes. Orestes admits that he killed his mother, and says that he did so following the commands of Apollo. He also says that his mother had it coming to her (because she killed Agamemnon, Orestes's father), and, unexpectedly, that she isn't related to him by blood anyhow. Now the Furies question Apollo. Apollo backs up everything that Orestes says, and elaborates on Orestes's point that his mother wasn't related to him. Apollo says that mothers are only incubators of embryos, and that only fathers are truly parents.
Shortly afterward, the trial proper wraps up, and the votes of the jurymen are tallied. While they are being counted, Athena announces that she's voting for Orestes, because she is biased in favor of men. As it turns out, the jury was split, but Athena's vote is the tie-breaker, meaning Orestes can go free… which is exactly what he does, pledging to make Argos and Athens allies forever.
The Furies aren't too pleased about this outcome, but after a lot of persuasion, Athena convinces them to accept it. At the same time, she convinces them to stay in Athens as goddesses in charge of helping good people and punishing bad people (instead of just punishing). From now on, instead of just being "Furies," the goddesses will be known as "The Kindly Ones," a.k.a. the Eumenides.