This is the second part of the Oresteia trilogy, written by Aeschylus. It takes place several years after 'Agamemnon'. Clytaemnestra and Aegisthus are still ruling, though some of the Thebans are still unhappy with the way they killed the old king.
The play begins with Orestes, the son of Clytaemnestra and Agamemnon, finding the grave of his father. Orestes has been away for years, in exile, and has returned for revenge. The murder of his father has weighed heavily on him, especially since the murderer has gone on, unpunished.
While at the grave, he runs into his sister Electra. She is bringing libations to pour on the grave. A libation is a drink offering given to the departed. Her mother, Clytaemnestra, is offering libations to try and assuage a bad dream that she thinks was sent from the underworld. She dreamed that she gave birth to a snake that then tried to suckle, causing a confusion of blood and milk.
The two meet and plan revenge. Orestes will pose as a messenger, bringing the news of his own death. He gets Aegisthus alone and then kills him. Then he summons Clytaemnestra so he can kill her too.
She is devastated by these events. She's just been told that her son was killed. Now she finds that this isn't true. Instead he has returned and killed her lover of many years. And now he means to kill her too. Her own son! She tried to talk him out of it, but he is resolute and slays her, across the body of Aegisthus, in the same spot that Agamemnon was killed.
Clytaemnestra is the most interesting in this play. Remember, she feels totally justified with how she dealt with Agamemnon. She then suffers a series of emotional shocks and is faced with her own death. At the end she pleads with her own son, her own blood, to spare her. But to no avail.
Orestes is also conflicted. He has been told by Apollo that this is justice but he is far from certain. If he could have figured a way out, he probably would have taken it but alas, there is no way. He ends the play by fleeing to a temple so that Apollo can wash the blood from him.
The 'Libation Bearers' doesn't have quite the punch of the other two plays, but I can see how it would be effective. Both Orestes and Clytaemnestra are in very tough situations without any apparent solution. The audience couldn't help but empathize.