Aristotle's famous book on drama called Poetics, gives a definition of what is required for a dramatic performance to be considered a tragedy. Enotes quotes his definition as:
Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech, with each of its elements [used] separately in the [various] parts [of the play]; [represented] by people acting and not by narration; accomplishing by means of pity and terror the catharsis of such emotions.
The upshot of this definition is that the play must have an emotional impact upon the audience, so that they are moved to pity and terror for the plight of the tragic hero. It is this empathy that provides the necessary catharsis or purging of emotions.
Aristotle also described the perfect tragic hero as one who has a tragic flaw that, a characteristic that is so developed in his nature, that it is the instrument of his tragic downfall.
In Macbeth, Macbeth himself is this tragic hero, and audiences can experience a catharsis at his downfall because, he began the play as a good and noble Thane, serving his king loyally, but, because of his flaw -- ambition -- he spirals into a killing machine who is unable to appreciate or find joy in life. He is finally killed by his enemies at the play's end, but it could be argued that he was already dead inside before the act of his physical death. His real death, death to his own humanness, comes from the relentless killing in service to a boundless ambition.
Below I have provided a link to more information on Poetics, and links to two essays which discuss Macbeth as a tragic hero in more detail.