Tantalos was a king in western Anatolia. For some now unknown reason, he killed his son Pelops and mixed the dead flesh into a banquet that he served to the Gods. Zeus restored Pelops' life and contrived an eternal punishment for Tantalos, but older powers than Zeus had been disturbed and the matter was far from over.
Military pressure forced Pelops out of Anatolia, and he led his followers to Greece, to the Peloponnese. There, Pelops murdered several people and, far worse, he broke an oath. One of his murders disturbed what we would call, "Nature" so much that it caused a famine all over the Peloponnese.
Pelops had two sons, Atreus and Thyestes. Atreus assumed the throne of Mycenae, but his wife, Aerope, became Thyestes' lover and tricked Atreus into surrendering the throne to him. But the Mycenaean people wanted no part of either Thyestes or Aerope, and Atreus was able to forcibly recover the throne. Atreus pretended to forgive his brother, and held a feast in his honor. Atreus waited until Thyestes found a cooked hand at the bottom of his plate, before he told Thyestes that he had been eating his own son.
Atreus had a son named Agamemnon, who became king of Mycenae and High King of the Akhaians when Atreus died, and it was Agamemnon who led the expedition against Troy. The Goddess Artemis refused to grant favorable winds for the expedition unless Agamemnon sacrificed his daughter, Iphigenia, to the Goddess. Agamemnon hesitated for agonizing months, while the army grew mutinous. He was torn between his loyalty to his daughter and his loyalty to his "spear brothers," he finally chose the latter. He sent word to his wife, Klytemnestra, to bring Iphigenia to the camp, claiming that she was to be married to the most esteemed of all the Akhaians, Achilleus. Klytemnestra joyously brought her daughter, Iphigenia was sacrificed, and Agamemnon went to Troy.
Meanwhile, Thyestes had a surviving son, Aegisthos. Aegisthos became Klytemnestra's lover, and together they planned to overthrow Agamemnon. We have several versions of what happened next, but the essential points are common to all and in no version do any of the story's figures do anything out of character. Everyone agrees that Klytemnestra and Aegisthos are a couple and act with one mind. Everyone agrees that Aegisthos is not outstandingly brave. Everyone agrees that Klytemnestra either tries to murder her and Agamemnon's son, Orestes, or exiles him, deprives him of his birthright and puts her boyfriend in his rightful place. Everyone also agrees that Agamemnon is not fought but murdered while unable to defend himself, and that Klytemnestra and Aegisthos then take over the throne. Agamemnon's subjects want nothing to do with this, but Orestes is only ten years old when this happens and they have no choice but to accept it.
Orestes grows up. He long hesitates to take a violent revenge that involves his mother, but the God Apollo orders him to do so and he does. As he is taking this revenge he, either voluntarily or involuntarily, according to which version you are following, kills his mother. It doesn't matter whether the murder is voluntary or not, and arguments as to its justice or lack thereof are a waste of time. Orestes' real crime was not against justice but against Nature, and just as Zeus' dispensation of justice to Tantalos did not solve anything, now this, the last in a series of crimes against Nature, was not to be touched by mere justice.
Orestes is finally purified of his guilt after a very long process involving many stages. At one point he bites off one of his fingers; this helps. At one point there is a trial. The logical arguments in it make only a limited amount of sense, but the trial is seen as a ritual, uniquely based on logic rather than song, and designed to limit and control the Furies of guilt, hatred and revenge such as those that had stalked the house of Tantalos for so long. Orestes is found not guilty and this helps, but it is only one step in the long process of quieting the Furies that swarmed in Orestes' soul.
But they do become quiet at last. Orestes' sister, Electra, was very much Klytemnestra's daughter, and she and Orestes would surely have come into conflict if the Furies had not been brought to rest. Orestes also had a son, Tisamenos, who grew up to be a fine man and a worthy king. Tisamenos had to fight wars against both Dorians and Ionians, but he didn't have to fight his own family. At long last, someone had calmed the Furies that had been ripping at the family ever since Tantalos.
How? After Agamemnon is dead, Klytemnestra calls on the Furies to rest now, now that she has everything she wants. She has no conflicts and no questions, she feels self-righteous and vindicated and well-deserving of the success she has achieved, and above all she thinks she is right. She is of course insane, and the idea that the furies will stop now that she has what she wants is a joke. Orestes knows he is wrong, he is completely clear on that point, and he will remain wrong whether he kills or not. His decision will bring clear-mindedness and integrity, but nothing will make him right, nothing will bring him any sort of satisfaction at all.
The trial is the climax of the Oresteia, because it makes a political point that doesn't mean much to us now. But in the traditional story the things said at the trial mean little, all that matters is that he stood it. He was the first person in his family to accept a trial, he was the first person in his family to fully accept his guilt, and he was the first person in his family that did not seek his own advantage. Achilleus was heroic for accepting his death and the time of its coming, Orestes was heroic for accepting the Furies. Aeschylos gets his character very well, he has Orestes say, "Let me kill her, then let me die." This and nothing else broke the chain that had destroyed the House of Tantalos. The Greeks universally regarded Klytemnestra and Aegisthos as the negative figures in the story, not because they were more wrong or evil than anyone else but because they were mediocre.