Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Initial Situation
Orestes is in exile.
Even though the play begins a few years after the end of Agamemnon (Part 1 of the Oresteia trilogy), the situation is much the same. Clytemnestra and her lover Aegisthus, Agamemnon's murderers, are still in power in Argos; Orestes, Clytemnestra and Agamemnon's son, is still in exile; and Agamemnon is still dead.
Conflict
Orestes comes home to Argos.
The major change in the status quo that has prevailed in Argos for the last who knows how many years is when Orestes comes home, ready for some revenge.
Complication
Electra and the Chorus of slave women come to offer libations to the tomb of Agamemnon.
Orestes realizes that he is not alone in his quest for revenge. This becomes clear once he hears Electra, as well as the palace slave-women, reveal that they, too, harbor deep resentments against Agamemnon's killers.
Climax
Electra and the Chorus offer Orestes their help.
Even though Electra doesn't end up helping Orestes in a concrete way, the Chorus will become a decisive ally. With these helpers (plus his friend Pylades), Orestes can now start seriously putting his plan into action.
Suspense
Orestes and Pylades approach the palace disguised as Parnassians.
Our valiant hero and his loyal sidekick are heading into the hornets' nest of their enemies. Will their enemies see through their disguises and phony Parnassian accents? Will they buy our heroes' story about how "Orestes" died under mysterious circumstances? Only time will tell.
Denouement
Orestes kills Aegisthus and Clytemnestra.
With the help of the Chorus, who make him come without bodyguards, Orestes is able to corner Aegisthus and kill him. Next, when he confronts his mother, he hesitates; then he gets a boost of encouragement from Pylades and ends up driving her into the palace to be killed. This marks the end of his revenge project.
Conclusion
Orestes goes crazy and heads for Delphi.
The conclusion begins when Orestes exhibits the bodies of Clytemnestra and Aegisthus, which proves beyond a doubt that his revenge is over. This very same action, however, brings him face to face with the enormity of what he has done, and makes him lose his mind. He runs off to Apollo's oracle at Delphi to be purified. If this sounds like a pretty inconclusive conclusion, you're right: Aeschylus had to make room for a sequel.