Oedipus the King has all the major conflicts: man vs. man; man vs. nature; man vs. god(s); man vs. fate; and--most importantly--man vs. himself.
The play is a kind of trial, with Oedipus playing all the parts: he is a prosecutor, a defendant, a judge and jury.
Here are some quotes by Oedipus to support each conflict:
MAN V. GODS: "Well argued; but no living man can hope To force the gods to speak against their will."
MAN V. MAN: "Monster! thy silence would incense a flint. Will nothing loose thy tongue? Can nothing melt thee, Or shake thy dogged taciturnity?"
MAN V. NATURE: "What plague infects our city; and we turn To thee, O seer, our one defense and shield."
MAN V. FATE: "I reck not how Fate deals with me But my unhappy children--for my sons Be not concerned, O Creon, they are men, And for themselves, where'er they be, can fend."
MAN V. HIMSELF: "he monstrous offspring of a womb defiled, Co-mate of him who gendered me, and child. Was ever man before afflicted thus, Like Oedipus."
I think that entire story is fraught with conflicts. Indeed, as previously noted, the conflict between Oedipus' free will and the fated destiny that was laid for him represents one of the most foundational conflicts present. Oedipus feels that he has to do battle with these forces, and challenge his fate through the use of his freedom. Another level of conflict present is the one where Oedipus the person must be challenged by Oedipus the ruler. When he has to find the source of the plague of Thebes, there is a conflict present. On one hand, Oedipus the person could very well wish that the cause is not discovered, but Oedipus the leader has staked his entire political capital on finding the cause. Finally, there is a conflict in Jocasta, who is the mother of what turns out to be her husband. There is conflict there.
There are different ways to answer this question.
You could say, for example, that the main conflict is between Teiresias and Oedipus. In this interpretation, the conflict arises because the prophet says Oedipus is guilty of killing King Laius and Oedipus believes he is not guilty.
Others would say that the main conflict is between Oedipus and his fate. In this interpretation, Oedipus is struggling to exert control of his own life.
Finally, some would say the main conflict is between Oedipus and himself. These people would argue that Oedipus is having to fight against his own impulses and his own arrogance.
A very good - and extremely difficult - question.
Jamie_Wheeler's response is a good synopsis of the background. But the play itself is about how Oedipus discovers this background. Oedipus' birth, upbringing in the household of Polybus and Merope, his flight from Corinth, and his fateful arrival at Thebes are all in the past when the play begins.
Oedipus' first objective is to rid Thebes of the plague by exposing and exiling the murderer of King Laius. But his search for the murderer is interrupted by the news of Polybus' death. Oedipus then begins his quest to discover the identity of his parents. In the end, of course, both the murderer and his parents are revealed.
So the "conflict" in the traditional protagonist/antagonist sense is unclear. Is Oedpius pitted against fate? Is it his unconscionable hubris that propels him to his doom? I'm afraid that there's no easy answer.
I would argue that the central conflict is that of man vs. himself. Oedipus must battle his own impetuousness, hubris, and consequences of his actions.
To understand the ramifications, a reminder of how Oedipus came to be king at all is necessary.
Oedipus' parents, Lauis and Jocasta, have been warned by an oracle that their newborn son is destined to marry his mother and kill his father. In an effort to avoid their fate, the baby is abandoned on a hillside and left to die. He does not die, but is found and grows into manhood, completely ignorant of his real parentage.
As an adult, Oedipus meets with Laius on the road. The two fight and Laius is killed. Because he does not know Jocasta is his mother, Oedipus ends up marrying her and inheriting the kingdom. Jocasta, too, is unaware that Oedipus is her son. Their children, therefore, are the products of incest. The oracle is fulfilled despite the human attempts to foil the Gods.
Teiresias, the blind prophet, tries to tell the hubristic Oedipus that his life is not what he thinks it to be. He warns, "But I say that you, with both your eyes, are blind:/You cannot see the wretchedness of your life,/Nor in whose house you live, no, nor with whom."
When Oedipus finds out the truth, he blinds himself, thus fulfilling Teiresias' prophecy both literally and physically.
man vs himself is an important theme in the play as well as the struggle of fate versus anyone else. In Oedipus, he tries his hardest to escape the harsh fate given to him by the Oracle. While moving to another Kingdom, unknowingly, he murders his father and marries his mother. He does anything to rid himself and in the end becomes the cause to his own demise. When he realizes his children and his brother/sister-children, he laments the fact that this could happen to him. He does not want to see anymore of what has become of him and blinds himself in the end.