Aristotle's definition of tragedy is that it revolves around "an action which has serious implications, is complete, and possesses magnitude." Thus, classical tragedies often take place in the world of kings, queens and princes. However devastating, a story of insufficient scope or scale may well fall short of tragedy.
In "The Godfather" the stakes are high on a number of levels. On the material level, there is a struggle for control of New York's crime syndicate. On the moral level, there is Michael Corleone's struggle to maintain his integrity and his family even as he rises to the top of the family business.
An authentic tragic hero is a complicated character whose creation requires considerable dramatic skill. He is neither a clear hero nor a villain. He is a flawed character who means well, but whose actions, due to a fatal blind spot, bring more harm and good. The tragic hero is a study in contradictions.
Michael Corleone, played in "The Godfather" by Al Pacino, is the one son of the Godfather intent on staying away from the family business. Following an assassination attempt on his father, Michael plays a critical role in saving his father and defending the family. While it is Michael's positive quality of loyalty that initially leads to his return to the family business, his pride and stubbornness drive him to be an even more ruthless leader than his father was.
The tragic hero sets out to do good, but his tragic flaws, often involving pride, conspire to undermine his intentions, and in the end his actions bring pain and suffering. In "The Godfather," Michael Corleone at first tries to stay clear of the family business. When he is drawn back into the business and eventually rises to the head of the family to become the new Godfather, Michael declares his intention to clean up the business and make it legitimate. But pride and ambition and paranoia lead Michael to make a series of ruthless decisions. A man motivated by loyalty to family ends up tearing his own family apart, murdering his brother-in-law and later his brother, and forever alienating his wife.
Catharsis is an elusive but essential aspect of tragic storytelling, and is the end result of the other dramatic elements discussed here. Introduced to a story of sufficient scale and sweep, the audience identifies with a character who means well, but whose fatal flaws only bring on pain and suffering. Pushed to a dramatic extreme, the audience experiences a vicarious purging or cleansing, and perhaps a kind of wisdom.