According to Aristotle, there are three elements that make a story a tragedy. The three elements (from the Greek) are hamartia, peripeteia, and anagnorisis, and all are present in Shakespeare's Macbeth.
Regarding hamartia, it is recognized that the "hero" has a tragic flaw; and the character's actions create problems that were not anticipated. In terms of Macbeth, Macbeth's tragic flaw is "vaulting ambition," or ambition that cannot be stopped; rather it trips over itself so that the character can, in this case, move up in the world.
I have no spur
To prick the sides of my intent, but only
Vaulting ambition, which o'erleaps itself,
And falls on th'other. . . . (I.vii.25-28)
Peripeteia indicates a change or "reversal" in one's circumstances.
...a sudden turn of events or an unexpected reversal, especially in a literary work, or...
[a] sudden and violent change in circumstances, especially in drama
In Macbeth, we might look to the witches' predictions for this reversal, or in terms of a violent change, we could refer to Macbeth killing Duncan and becoming a murderer—and then King.
The last element of tragedy, according to Aristotle is when the main character has something like an epiphany: when a critical piece of information is revealed to him (her). Anagnorisis...
...is a moment in a play or other work when a character makes a critical discovery... or,
Anagnorisis is the recognition by the tragic hero of some truth about his or her identity or action that accompanies the reversal of the situation in the plot, the peripeteia...
In this case, it occurs when Macbeth finally realizes that the witches' have led him to his doom, when all of the seemingly impossible "caveats" in their predictions actually occur. It looks like Birnham Wood is moving to Dunsinane Hill.
Amid these three elements, Macbeth is also a tragic hero. Aristotle defines the tragic hero as a "great" (important) man; he dies due to his tragic flaw; and, his death is his own fault. Macbeth is a great man: he is a decorated and honored soldier of King Duncan, well-loved by the King and admired by his peers. He has fought like a lion on the battlefield for his King. Macbeth dies because of his tragic flaw: vaulting ambition. He tries to be happy with all that Duncan has given him—the rewards and the honors. He tells Lady Macbeth he does not want to continue in their plot to kill the King, who Macbeth truly loves and admires. Lady Macbeth insults his manhood, and Macbeth gives in to her nagging and his ambitious nature, and ultimately, he dies. His death is his own fault because he does not ignore the witches' predictions—as does Banquo—and he gives in to his wife's desires, allowing her to manipulate him when he really does not want to commit regicide; however, she wants to be queen.
And so, Macbeth is a tragedy by Aristotle's definition, and Macbeth a tragic hero as well.