Do you mean the clash between Egypt and Rome in the play?
In the play, Antony and Cleopatra, Mark Antony is one of three rulers of Rome (Triumvir). Egypt is part of the Roman empire (do you remember that the Roman empire was HUGE at one time in history?) Mark Antony, who was in love with Cleopatra of Egypt, left Rome and went to Egypt to be with Cleopatra in spite of the fact that Octavius (another Triumvir) wanted him to stay and help him fight pirates.
Antony declares Cleopatra and himself king and queen of Egypt. He feels that Octavius has not given him his fair share of land as one of the three rulers of Rome. This angers Octavius, so Antony (now of Egypt) and Octavius wage a huge sea battle against each other. I believe this is the clash to which you are referring.
Cleopatra sends her navy to help, but right in the middle of the fight, she calls her navy back to Egypt. Mark Antony accuses Cleopatra of betraying him, she tries to win him back by pretending to kill herself hoping that he will rush back to Egypt to see her, dead. Rather, he decides he cannot live without her and asks his servant to kill him, but the servant cannot do so, and Antony tries to kill himself, but fails (sounds like Romeo and Juliet, huh?).
Anyway, with Antony now dead, Octavius tries to convince Cleopatra to surrender, but she does not want to be taken prisoner and paraded through Rome (this is what the Romans did after every victory - remember the play Julius Caesar?) so this time, she really kills herself.
The first clash between Rome and Egypt involves Antony AND Cleopatra against Octavius, and when Antony dies, it involves Cleopatra against Octavius.
Read about the play here on enotes. Read the text at the link below.