Identify and explain the major internal and external conflicts in Oedipus the King.

In Oedipus Rex, tragic hero Oedipus engages in several conflicts with other characters, partly as a result of his tragic flaws: his short temper and arrogance. When Teiresias comes to Thebes at Oedipus's request but doesn't want to reveal the harrowing truth of Oedipus's background, Oedipus quickly becomes angry and lashes out at the prophet. This leads Teiresias to return Oedipus's insults and to tell him what he's done and what will happen as a result. Oedipus's paranoia leads him to suspect his brother-in-law, Creon, of conspiring with the blind prophet to oust Oedipus from power. A final external conflict takes place before the start of the play, when Oedipus kills the king and his attendants after they run him off the road to Thebes. The king was Laius, his biological father, and Oedipus's killing of him sets the prophecy into motion.
The play also depicts Oedipus's internal conflict as he discovers and struggles to accept the truth of his past and its effect on his present. He is in denial for much of the play, hoping that he did not kill his father and marry his mother. Eventually, the truth becomes clear, and he can no longer deny it. He sees that his wife/mother has killed herself and gouges his own eyes out. His internal struggle has become externalized, as he now serves as a symbol of the consequences of mental blindness. He also becomes an example of the quintessential fall from grace: he goes from the highest point of power to the lowest state, such that even the common people of Thebes no longer envy his position.

In addition to the conflicts listed by the other educator, Oedipus also conflicts with his wife, and mother, Jocasta, during the course of his search for Laius's killer and the truth about himself. Jocasta becomes anxious about Oedipus, trying to reassure him that his prophecy—that he will kill his father and marry his mother—cannot come to fruition. When a messenger comes, bearing the news that Oedipus's father in Corinth, Polybus, is dead, Oedipus feels that the part of the prophecy that concerns his mother could still come true. Jocasta, trying to convince him otherwise, says,

Why should a mortal man, the sport of chance,With no assured foreknowledge, be afraid?Best live a careless life from hand to mouth.This wedlock with thy mother fear not thou.How oft it chances that in dreams a manHas wed his mother! He who least regardsSuch brainsick phantasies lives most at ease.

But Oedipus will not be comforted. When the messenger tells Oedipus that Polybus and Merope are not his biological parents, but that he, himself, received the baby Oedipus from a servant in Laius's house, Oedipus burns to know his true parentage. However, Jocasta, fearing the truth begs him,

Oh, as thou carest for thy life, give o'erThis quest. Enough the anguish I endure.

Oedipus, however, assumes that Jocasta's "pride of ancestry" makes her fear that she has actually married the son of a servant rather than a prince. She begs again, telling him it is actually for his own sake that she asks him to stop searching. When he refuses her, she leaves the stage and takes her own life (having, evidently, figured out the truth).

Oedipus experiences several conflicts over the course of the play. Some occur in the period in which the play takes place, although a crucial one occurs before the beginning of the play and is responsible for many of the events that occur during the play. The major external conflicts involve the following characters.
Laius: Before the play starts, Oedipus has killed Laius. This makes his death patricide and (along with his marriage to Jocasta) is the cause of the plague.
Creon: Oedipus suspects that Creon wants the throne. Oedipus is more of an individualist and Creon more of a traditionalist; Oedipus often disagrees with Creon on the issue of respect for tradition, custom, and prophecy.
Tiresias: Oedipus presses Tiresias to speak despite the prophet's obvious reluctance. Oedipus also gets angry when Tiresias says things he does not wish to hear. 
The major internal conflict occurs when Oedipus realizes what he has inadvertently done. He struggles with how to live with the burden of this knowledge and eventually blinds and exiles himself rather than continue to pollute Thebes.