Electra opens in Mycenae, in Ancient Greece. Before we can delve into this story, we need a bit of back story. When Agamemnon, the resident King, returned from the Trojan War, his wife Clytemnestra and her lover Aegisthus murdered him and usurped the throne. The daughter of Agamemnon and Clytemnestra, Electra, smuggled her brother Orestes out of town so that he could grow up safe in a foreign land and come back some day to avenge their dad.
And that day is now. Orestes is back in town and he is ready for revenge. He brings with him the old slave who smuggled him out of the palace the day Agamemnon was murdered. In addition, Orestes brings his buddy Pylades who, as far as we can tell, will be like to a prop for the rest of this play. Orestes hatches a plan with these men: the old slave will go to the palace and pretend to be a messenger from Phocis bringing news of Orestes's death. Meanwhile, Orestes and Pylades will go pay their respects to Agamemnon's grave.
Now we jump to Electra – Orestes's sister who stayed behind in Mycenae after Agamemnon was murdered. Because she refuses to embrace the new king and because she won't forgive her mother for murdering her father, she's abused on a regular basis by the royal couple. She spends all day grieving and praying that Orestes will soon return to Mycenae to right these royal wrongs. In contrast, her sister, Chrysothemis, kisses up to Clytemnestra and Aegisthus and lives a life of comfort. The two sisters argue extensively about whose course of action is better. Electra maintains that they have a filial duty to see their father's murderers brought to justice, while Chrysothemis pragmatically decides that there's nothing to be gained from snubbing the King and Queen.
Next we get an intense scene between Electra and her mother, Queen Clytemnestra. Electra calls her a murdering sinner, but the Queen has her own side of the story. Agamemnon sacrificed their daughter Iphigenia in order to get his ships home from the Trojan War. That Agamemnon sacrificed Iphigenia infuriates the Queen. In her mind, her husband deserved to die. Electra, however, is having none of it. Before the argument gets too catty, the old slave shows up with what we know to be fake news: Orestes is dead. Electra is beside herself with grief, but the Queen is relieved. Queen Clytemnestra no longer has to worry about her son coming back to kill her. She takes the slave inside to celebrate.
Electra grieves alone outside the palace and resolves that, now that she can't rely on her brother, she has to kill her mother herself. Before she starts sharpening her sword, Orestes shows up with his friend, Pylades. When he realizes that this bereaved woman is his sister, he reveals his true identity. Much embracing and rejoicing follows. The two of them quickly agree to avenge their father together. By "together" they mean that Orestes will go inside and stab Clytemnestra, while Electra stands outside and encourage him verbally. So they do. We know from Clytemnestra's screams that the deed is done.
Orestes comes back outside, carrying Clytemnestra's body covered in a sheet. Electra is overjoyed. Just then King Aegisthus shows up, pleased as punch because he heard that Orestes is dead. (He's about five minutes behind everybody else.) Yep, says Orestes, and here's his body! It doesn't take long for Aegisthus to realize that the dead body is his wife and the man before him is Orestes. He resigns himself to his fate (being murdered by Orestes), and is ushered inside the palace as the play comes to a close.