Act I:
The play begins with a group of citizens who have come to the palace of King Oedipus of Thebes to ask for his help. The city is being ravaged by a plague, and the people are looking to their king to provide a solution. Oedipus promises to do all he can to save his people and has already sent his brother-in-law, Creon, to the oracle at Delphi to ask for guidance. Creon returns with the news that the city must find and punish the killer of the former king, Laius, in order to end the plague. Oedipus declares that he will find the murderer and punish him, no matter who he may be.
Act II: Oedipus begins his investigation by questioning the blind prophet Tiresias. Initially reluctant to speak, Tiresias eventually tells Oedipus that he himself is the killer of Laius, and that he is living with his own mother. Oedipus refuses to believe this and accuses Tiresias of being in league with Creon to overthrow him.
Act III: Creon returns and denies the accusations, but Oedipus is still suspicious. Jocasta, Oedipus's wife, enters and tries to calm him down by telling him the story of how Laius was killed. As she tells the story, Oedipus realizes that he was the man who killed Laius, and that Laius was his father. Shocked and horrified, Oedipus leaves the stage.
Act IV: Jocasta is left alone on stage and begins to realize the terrible truth of her situation. She remembers how she and Laius abandoned their son on a mountainside when he was born, and she begins to fear that Oedipus may be that son. A messenger enters and confirms her fears, telling her that he himself had rescued a baby from the mountains and given it to the childless king and queen of Corinth, who had raised the baby as their own son - Oedipus.
Act V: Oedipus enters the stage, his eyes gouged out, having just learned the truth of his identity and having blinded himself with his mother/wife Jocasta's brooch. He begs Creon to exile him from Thebes, and Creon agrees, promising to care for Oedipus's two daughters, Antigone and Ismene. The play ends with the chorus commenting on the irony of Oedipus's fate - that the man who sought to save his people has ended up destroying himself and his family.
That's a brief summary of Oedipus Rex. The play is a tragic exploration of fate, free will, and the consequences of our actions.