She is the daughter of Agamemnon, the king of Mycenae, and Clytemnestra. Electra's name refers to fire or spark and is also associated with amber.
Electra rescued her younger brother Orestes when her mother murdered her father on his return from the Trojan War. Although Agamemnon was no angel and had offered his other daughter Iphiginia to Artemis for fair winds, Clytemnestra had a public affair with Aegistus while her husband was away and was thus no angel either.
Later, when Orestes returned as a grown man, they plotted against their mother in revenge and Electra gave moral support to her brother to commit the evil dead. When the Furies, the deities who wreak vengeance on murderers, pursued him, she was overwhelmed with remorse.
Freud named the Electra Complex (feminine equivalent of the Oedipus Complex) after her because of her attachement to her father and hatred for her mother.