How can hubris lead to a character's downfall?
The term "hubris" in most literature classes refers to a particular element of the analysis of Greek tragedy in Aristotle's Poetics. According to this account, the typical protagonist or tragic hero is a noble character, greater and more powerful than the average person, whose downfall is occasioned by hubris. For us to feel "fear and pity" at this downfall, the reversal of fortune must be caused by a combination of external events and the character's own weaknesses or "tragic flaws."
Hubris, in this context, means overbearing arrogance. Often it is framed as a challenge to the authority of the gods, in which a mortal forgets that no matter how powerful or wise he may be, he can never triumph against fate or the will of the gods. Often the protagonist is portrayed as receiving warnings from prophets or natural signs that the gods disapprove of his actions, and nevertheless persists in those actions, and then is punished by the gods.
For example, in Sophocles' Antigone, Creon decides to punish Antigone and leave the corpse of Polynices unburied, despite evidence of divine preference and the warnings of Tiresias. As a consequence, Creon's wife and son commit suicide.
The mechanism by which hubris leads to downfall is usually fate or divine intervention, but sometimes can be the inherent logic of the protagonist's actions.