Octavius (a.k.a. "Young Octavius") is Julius Caesar's adopted son. Like his adoptive father, Octavius doesn't appear on stage that much. Throughout most of the play, Octavius is off travelling the world. He returns to Rome when Caesar is assassinated and joins forces with Antony against the conspirators.
Octavius may be "young" (Antony likes to remind him that he is older and more experienced), but he's not a pushover. Check out what happens when Antony tries to take command during the battle at Philippi.
Octavius, lead your battle softly on,
Upon the left hand of the even field.
Upon the right hand I; keep thou the left.
Why do you cross me in this exigent? (5.1.2)
This minor game of tug-of-war between Octavius and Antony foreshadows what will happen between the two men after they defeat the conspirators. Although not portrayed in this play, Antony and Octavius (along with Lepidus) will go on to rule Rome as part of the "Second Triumvirate" (the first included Caesar and Pompey)
As every student of history (and Shakespeare) knows, that whole "let's rule Rome together" thing didn't quite work out. You can read all about this in our summary of Antony and Cleopatra.