The overarching conflict of the play is, as others have already mentioned, that between Oedipus and himself. But in the course of this conflict playing out, a number of other important conflicts are generated as a result. Oedipus's overriding desire to find Laius' murderer (himself) leads him into direct conflict with the blind seer-prophet Tiresias.

At first, Tiresias is reluctant to reveal the identity of Laius's killer—not surprisingly, when you consider that he's actually standing right in front of him. Tiresias's reluctance to divulge this crucial information leads Oedipus to accuse him of being the murderer. Tiresias is then left with no choice: he must tell the truth.

But Oedipus simply cannot believe his ears. And so we're introduced to another conflict related to Oedipus vs himself: Oedipus vs the truth. Oedipus has pledged to get at the truth, to find out who killed Laius no matter what, but when all the sordid details of what really happened are revealed to him, he instantly goes into denial. Oedipus is no longer concerned with the truth, but with avoiding its terrible consequences. As soon as Tiresias speaks those fateful words, he rightly senses that the truth will eventually destroy him.

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.

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."

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.

What is the major conflict in Sophocles' Oedipus Rex?

The most predominant conflict in Oedipus Rex is character vs. fate. Oedipus was born to a cursed family and was therefore cursed himself. According to myth, King Laius was King Pelops' charge. Laius became such close friends with Pelops' youngest son Chrisippus that they ran away together. As a result, Pelops cursed Laius (Tripp, Meridian Handbook of Classical Mythology). Hence, it is as a result of Pelops' curse that Oedipus is born to the fate of killing his own father, Laius, and sleeping with his own mother. Therefore, the most predominant conflict reflects circumstances far beyond Oedipus's control, making the predominant conflict character vs. fate.

However, this would not be a true Greek tragedy, particularly not the tragedy Aristotle lauds in his Poetics as the most perfect tragedy, if Oedipus Rex did not also possess a character flaw that helps propel him towards his great fall (Barbara F. McManus, "Outline of Aristotle's Theory"). Oedipus's character flaw is recognized to be his excessive pride, making the second predominant conflict in the play character vs. self. We particularly see evidence of Oedipus's pride in his dealings with both Creon and Tiresias. It is Oedipus's pride that drives him to suspect a plot to overthrow, not only Laius, but himself and to disbelieve Tiresias's prophecy. Instead, Oedipus believes that both Creon and Tiresias are conspiring against him and that Creon paid Tiresias to deliver false prophecy, as we see in Oedipus's lines:

... the trusted Creon, my friend from the beginning, beguiles me and secretly desires to oust me, engaging this craftily-working wizard ... who sees clearly only for profit. (405-409)

It is also Oedipus's pride and even his impetuous temper that leads him to kill a man who ran him off the road at the intersection heading towards Delphi--a man that turned out to be his own father. Therefore, the second most predominant conflict in the play is character vs. self.

What is the major conflict in Sophocles' Oedipus Rex?

The major conflict in this play is between reality and the desperate attempts of Oedipus and some of the other characters, most notably Jocasta, to ignore the facts that point to one inevitable conclusion: that Oedipus is the man responsible for the death of the former king of Thebes, and therefore the man responsible for the plague that is troubling Thebes. Note what Oedipus says in the following quote when his suspicions about his own identity increase massively:

Oh, but if there is any blood-tie

between Laius and this stranger...

what man alive more miserable than I?

More hated by the gods?

Of course, the play gradually presents both the audience and Oedipus with evidence that becomes more and more unavoidable, pointing the finger directly at Oedipus and forcing him finally to face the facts about his own identity, with tragic results. Reality wins out. The play thus places the forces of reality against the forces of intentional ignorance, or trying to ignore the facts for as long as possible. Both Jocasta and Oedipus finally have to accept the truth, and this brings disaster into their lives.

What philosophical issues does Oedipus face in Oedipus the King?

Oedipus is faced with philosophical issues concerning fate.

Fate is sometimes a confusing subject in regards to its role in a story. In Western storytelling it's often seen as a binary good-or-bad force with mysterious origins and powers. The "good" facet is often used in relation with words like "destiny", and the "bad" aspect in relation to words like "doom". Fate is often seen as something to be embraced if it's good, or avoided if it's bad. Many modern stories make use of the trope wherein a character "escapes their fate" through force of will or by undertaking specific tasks (a good example would be Luke Skywalker's rejection of his "destiny" to turn to the Dark Side).

This makes fate out to be something one can disagree with, like fortune-telling, or something that possesses a personality and decision-making of its own, but is only slightly more powerful than the average person since it can be "defeated" with enough skill, luck and guidance. So, fate is often employed as a "man vs. nature" or "man vs. himself" conflict.

While elements of these conflicts are apparent in Oedipus, the moral of the story is more definitive; it is a "man vs. God" conflict, one which the man can't win. Fate, in this play, and in much of Greek tradition, is not something that can be altered or rejected. It is a course set by the universe itself, something that not even the gods can alter (in fact the gods were known to be less powerful than the Fates and subject to their decisions).

Oedipus faces several issues;

1. Should he consider the prophecy to be fate? Just because someone says "X will/won't happen" doesn't mean it's an actual prophecy or fate, but this message came from the oracle at Delphi; it would be hard to find a higher authority on Earth.
2. Should he attempt to reject his fate? No matter the reason, this would imply that he thinks he can defy the universe, a pride which cannot go unpunished in itself.
3. If he rejects it, in what manner does he do so? A wise character might be lauded if their defiance is undertaken for noble causes, such as protection of the innocent, whereas a character rejecting fate for the purposes of their own satisfaction and fortune will be cast as foolish and deserving of punishment.
4. If he accepts it, does his humanity diminish? It might be argued that any truly "human" character will reject fate because they don't really know what's best for them (note that the Greek concept of free will, if it existed, didn't match our own, and that what we call free will was more likely to be interpreted as a cause driven by some force other than divinity).

Oedipus learned of his fate upon visiting the oracle, and his response was to reject it by running away. He did so in order to protect those that he loved. It seems that it never occurred to Oedipus that his "parents" might not in fact be his birth parents, nor did the oracle choose to specify this. There's no way of knowing if Oedipus's decision to run away is what directly led to the murder of Laius, or if Laius would have found his way to death at Oedipus's hands even if Oedipus had stayed home. This ambiguity is what lends fate its power in this story.

The redeeming element might be that Oedipus is not the prime actor in this prophecy. Laius and Jocasta were initially given this prophecy, and their abandonment of Oedipus as an infant was, in fact, the first rejection of the prophecy. Oedipus rejected the prophecy out of compassion for others, whereas Laius and Jocasta rejected it out of fear for their own lives. Since Oedipus ultimately accepts his fate, I think he can be considered a tragic hero rather than someone entirely "deserving" of his fate.

What philosophical issues does Oedipus face in Oedipus the King?

The philosophical debate between free will and determinism is also relevant here. Do we have the freedom to make our own decisions and choices in life or are they determined for us in advance? The question is of particular importance in relation to Oedipus because he lives in a time when just about everyone believes that the gods regularly intervene in human affairs. By the standards of the day Oedipus is certainly guilty of hubris, or overweening pride, in attempting to defy the prophecies of the Delphic Oracle. The implication here is that everything that happens in our lives is determined by fate, and that any attempt to defy that fate will inevitably end in disaster.

Yet at the same time, Oedipus is only human. He is a king, a very important individual who needs to impose himself on the world around him. Of necessity he must be a dynamic character; he must make things happen rather than allow things to happen to him. He can't just go through life passively accepting everything that happens, even if he wants to.

But this tragedy of fate is a tragedy for us all, a tragedy of the human condition itself. Even if we don't believe in fate, almost all of us will, at some point in our lives, try to exceed our grasp; to step outside ourselves, as it were, to attempt something great that will earn us praise and honor in other people's eyes. This is what Oedipus does when he solves the riddle of the Sphinx. And it is this act of hubris, this very human act, which ultimately precipitates his tragic downfall. 

What philosophical issues does Oedipus face in Oedipus the King?

Another philosophical issue Oedipus must deal with has to do with truth. Is it better to know the truth about oneself, another person, or anything, if that knowledge is going to make one unhappy? Is it better to live in blissful ignorance? Certainly, Oedipus was a much happier person before he learned he murdered his own father, married his mother, and fathered children with the same woman who bore him.  The knowledge that both he and his father "planted their seeds" in the same maternal "soil" is terribly upsetting for Oedipus (and for his mother, who kills herself after learning the truth). 

Would it have been better for the truth to remain hidden and thus spare the feelings of Jocasta, Oedipus, and their children? It seems that, for Oedipus at least, the answer is "no." Thebes has fallen victim to a blight on the crops, a decline in birthrate, and so forth, as a result of Laius's murderer having gone unpunished (according to the oracle). In learning the truth, Oedipus effectively solves Thebes' problems. His ruin, then, permits others to live better lives.  It would have been personally better for Oedipus, however, had he never learned the truth. The old adage "be careful what you wish for" is certainly true for him in this case.

What philosophical issues does Oedipus face in Oedipus the King?

In Oedipus the King, the character of Oedipus has to face the weight of an overwhelming guilt that dooms him. Philosophically, Oedipus has to deal with the fact that while the actions that led to his guilt and ruination were his own, he had no idea that such a fate would befall him. Events spiraled out of his control, and he was powerless to stop them. Did he deserve such a cruel fate, or was the universe simply indifferent to his happiness?

Although Oedipus complains about his fate loudly and pitifully, he does in fact accept it. He is the murderer who inadvertently cursed Thebes, and he must be banished, by his own edict. Even though destiny has dealt him a terribly unfair hand, he acquiesces to the will of the gods and gives up everything, including his home, family, and throne.

What are the motives and conflicts of some of the characters in Sophocles' Oedipus Rex?

Creon falls into a conflict with Oedipus because Oedipus accuses him of treason. Oedipus believes that Creon has been plotting to take the throne since the reign of King Laius. He believes that Creon paid bandits to kill King Laius and now Creon is allowing Oedipus to be blamed for Laius's murder. In fact, Oedipus believes that Creon has paid Tiresias to deliver false prophecy blaming Oedipus of Laius's murder as well as the horrible prophecy that he will realize he has had children with his own mother. Hence, we can say that Creon's conflict is man vs. man. However, despite Oedipus's belief, Creon's motives are actually pure. Creon is actually loyal to the king and quite content with the power Oedipus has extended to him, as we see in his lines:

For now I have everything from you without fear; but if I myself were ruler, I'd do much against my will. How then could tyranny be sweeter to me than trouble-free rule and sovereignty? (615-619)

Hence, we see that Creon's conflict is man vs. man, but his motives for action are to heal the city, make peace with Oedipus, and live peacefully in the city under Oedipus's rule.

Jocasta is a character that has a conflict with fate and a conflict with herself as well, making her conflicts character vs. fate and character vs. self. Jocasta is a victim of fate. When she learned of the prophecy that her son would one day kill her husband and sleep with her, she thought she was taking measures to prevent the fulfillment of the prophecy by having her son killed. However, instead, the shepherd to whom she gave him took pity on the baby and gave him to someone from Corinth who gave the baby to King Polybus to raise. Because Oedipus continued to live, he fulfilled the prophecy even though Jocasta had believed for a long time that the fulfillment of the prophecy was impossible. Hence, Oedipus fulfilled both his and Jocasta's fate. Since Jocasta tried to escape her fate, we can say that one of her conflicts is with fate. However, she is also in conflict with herself because in the beginning of the play she believes as she has believed for years that she had escaped her fate. Regardless, as the play progresses, she hears enough evidence to learn the truth, which drives her to commit suicide, showing us that another one of her conflicts is character vs. self. Jocasta does everything she can to escape hearing the truth, even trying to convince Oedipus not to send for the shepherd. Her motive is to protect her husband from the truth, as well as herself.

What is the conflict in Sophocles' Oedipus Rex?

Two predominant conflicts in Oedipus Rex are Human against Fate and Human against Self. All through the play, Oedipus, in a Human against Fate conflict, is unwittingly chasing the very fate he is trying to escape. Indeed, Oedipus was meant to be left as dead as an infant for the purpose of outwitting and escaping fate. As Oedipus's actions in the end of the play and his final exile show, he wholly and completely failed to escape his fate in any way. Therefore, Sophocles makes a strong statement advocating the power of Fate over humankind.

O cloud of night,
Never to be turned away: night coming on,
I can not tell how: night like a shroud!

Oedipus struggles in an inward direction throughout the play against his hubris, which is defined as excessive, detrimental arrogance and pride. This hubris is his tragic flaw and the substance of his Human against Self conflict. It is Oedipus's hubris that leads him to require the disclosure of the fateful information held by Creon and Teiresias. It is, likewise, his hubris that leads him to swear to discover the identity of the murderer of King Laius and punish him fully. The conflict of Oedipus against self here, of course, is that Oedipus himself is the murder and the one whom he is sworn to discover and punish.

Again, it is clear in the ending of the play that Oedipus is wholly and completely incapable of coming out of this conflict victoriously, even though he has opportunity to step away from the conflict--he fails to do this because he fails to recognize there is a conflict, because he fails to recognize his hubris. Therefore, Sophocles makes a strong statement about our helplessness against our own inner flaws, about which we are blind--unless someone, like a Creon, can convince us to see rightly.

Then once more I must bring what is dark to light.
[...]
You shall see how I stand by you, as I should,
To avenge the city and the city's god.