Aristotle defines a tragic hero as a man who is great—meaning, in this case, a hero or someone highly regarded by others. Our hero must die, and his death is his own fault, generally brought about by his tragic flaw.
In Shakespeare's Macbeth, Macbeth is a great man: at the beginning of the play, Macbeth has just come from the battlefield, and King Duncan receives news of Macbeth's valor in the latest skirmish with the Norwegians. The Sergeant reports:
For brave Macbeth—well he deserves that name
Disdaining fortune, with his brandish'd steel...
carved out his passage
Till he faced the slave... (I.ii.18-19, 21-22)
Duncan is full of praise for Macbeth and heaps rewards upon him when they meet. Reading further, we find that Duncan is not only Macbeth's King, but also a cousin and Macbeth's friend. Duncan loves him like a son and pledges to continue rewarding him for all he has done in defense of his native Scotland.
A tragic hero must have a tragic flaw. Macbeth is quick to note that his flaw is "vaulting ambition:" ambition that is never satisfied. His desire to be greater—more powerful—is insatiable.
Finally, our tragic hero must die because of his flaw. Macbeth has listened to and believed the predictions of the three witches. However, they provide him only with "half-truths," things that aren't really accurate in the deepest sense—they sound plausible, if not totally credible. Macbeth accepts all that the witches tell him, even knowing they are creatures that serve evil. Hecate, queen of the witches, notes that man's "chiefest enemy" is a false sense of security.
He shall spurn fate, scorn death, and bear
His hopes ’bove wisdom, grace, and fear.
And you all know security
Is mortals’ chiefest enemy. (III.v.30-34)
The witches lure him on with their enticing lies until Macbeth realizes he has been tricked and has no way to turn back. For one last time, we see Macbeth's greatest as he chooses to fight to the death rather than surrender to Malcolm. (Macduff eventually kills him.) So Macbeth declares:
Yet I will try the last. Before my body
I throw my warlike shield! Lay on, Macduff,
And damn'd be him that first cries, “Hold, enough!” (V.viii.37-39)
Macduff enters and reports his victory to Malcolm:
Hail, King! for so thou art. Behold where stands
The usurper's cursed head. (63-64)
According to Aristotle's definition, Macbeth is a tragic hero.