Irony of Vengeance
When Clytemnestra kills Agamemnon for sacrificing their daughter Iphigenia, she believes she is righting a wrong that has been done to her, balancing the scales of justice. Ironically, she has actually unbalanced them again, causing another person (Orestes) to kill her out of revenge in the same way that she killed Agamemnon.
Irony of Electra
Clytemnestra kept her daughter Electra in the city in order to keep her on a tight leash, preventing her from working against her. Ironically, this backfires, as when Orestes returns, Electra is there to provide him with inside information about the best way to sneak into the palace to murder Clytemnestra.
Irony of the Eumenides
The Furies are notoriously furious goddesses (hence the name) who care more about vengeance than about objective justice, such as one might ideally find in a court of law. At the end, ironically, Athena makes them into guardian protectors of Athens, presiding over the city to ensure complete justice with regard to the law. This is a complete character shift, turning them into the Eumenides ("Kindly Ones") rather than the Furies.
Irony of Divine Commands
Since Orestes is acting on the command of Apollo, who is a powerful god, it seems that he shouldn't have to deal with negative repercussions for his actions - he didn't exactly have a choice. Ironically, however, the Furies grow angry at him and chase him all around Greece, and Apollo waits quite a long time before intervening on his behalf.
Irony of Athena's Sexism
Even though Athena is a woman, she herself plainly says that she prefers the man "in all things but marriage." This disposition toward men is the foremost reason she votes in favor of Orestes in his trial at the end, setting him free, which is an ironic ending in the context of the whole play.
