What is the signifigance of the curse on Oedipus in Oedipus Rex?

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The prophecy (or curse) that haunts Oedipus is significant on several levels. This prophecy effectively drives the plot of the play. Also, the prophecy defines the conflict facing the protagonist while also pointing to the symbolic and thematic nature of this conflict. 

The action of the story relates directly to the actions that Oedipus has taken to avoid the fulfillment of the prophecy (that he will murder his father and marry his mother). It is the prophecy that leads him to travel to Thebes, as he cannot passively accept such a horrendous fate. In attempting to evade his fate, Oedipus rejects the necessary truth of that fate. 

Oedipus struggles against the oracle that predicts his hand in his father's death and boldly asserts that it is wrong...

This struggle against predestination is the central thematic conflict of the play. Oedipus has been set againt the gods. 

Another theme in the play is the distinction between the truthfulness of oracles and prophecies of the gods (fate), as opposed to man's ability to influence his life's trajectory through his own actions...

The curse placed on Oedipus is too terrible for him to bear and so he rebels. His rebellion only serves to fufill his fate, creating a tragic irony. 

Though Oedipus is a strong-willed leader and a boldly honest man, he is not defined by these traits as much as he is defined by the situation created by his curse. In the end, he is defined exactly by the circumstances predicted by the prophecy. He has murdered his father and committed incest with his mother. 

Oedipus' powerlessness against fate is significant and can be seen as one of the play's final commentaries on the state of man and on free-will. Oedipus is cursed, ultimately, as all men are cursed, to fulfill his fate regardless of any agreement or disagreement on his part.