In Oedipus Rex (or Oedipus the King), the major conflict exists based upon what Oedipus has unknowingly done in his life to wreak the havoc (in the form of plagues) that now visits Thebes.
Before Oedipus was born, his fate was already decreed: he would kill his father and marry his mother. The gods are responsible for defining his actions before Oedipus takes his first breath.
To avoid the fate the gods have ordained for the King of Thebes (believing they can do so: the major theme in the story), Laius and Jocasta agree that the child she has delivered be killed. Oedipus is taken with instructions that he be disposed of, but the soldier has not the heart to do so. He abandons the baby, believing the infant will die. Oedipus is found and taken in by a shepherd, and ultimately finds his way into the palace of the King of Corinth, who adopts the baby and raises Oedipus as his own son. Oedipus has no knowledge of any of this.
When Oedipus crosses the path of Laius and his slave, they insult and attack Oedipus, without knowing who he truly is. Oedipus defends himself and kills Laius, thus (unknowingly) fulfilling the first part of the prophecy.
Oedipus later solves the riddle of the Sphinx who has terrorized the people of Thebes (and all travelers in the area) for a very long time; in doing so, Oedipus is hailed as a hero, and it is suggested that he marry Jocasta, Laius' widow. Oedipus agrees, once again unknowingly fulfilling the second part of the prophecy.
He and Jocasta have two children, Ismene and Antigone.
Years have passed, and plagues have befallen Thebes. The oracle ("fortune teller") explains that the cause of these calamities is the murder of Laius that has never been solved. Oedipus, a good and just king, orders that the culprit must be found and punished.
Herein, I believe, lies the main conflict of the story: the plagues must be stopped and the murderer found. However, the murderer is, unbeknownst to all of them, Oedipus. And Oedipus will have to own up to what he has done, if he is truly a just and noble man and king, although at first he does not believe what he hears.
Based upon Sophocles (the author) and his culture's system of beliefs (predestination), the moral of the story is that you cannot evade or escape your fate: it will catch up with you, and there is nothing you can do to stop it.