Who is Hamlet?

Hamlet, Prince of Denmark, is the title character of William Shakespeare's longest and arguably most influential play. The tragedy has been continuously retold for four centuries in nearly every corner of the world, with a seemingly endless ability to be endowed with new life and energy. 
So what makes this play—and this character—so staggeringly monumental?
The brief plot synopsis is as follows. Hamlet, Prince of Denmark, returns home to find his father dead and his mother Gertrude already remarried to his untrustworthy Uncle Claudius. Suspecting foul play, Hamlet's suspicions are confirmed when his father's ghost visits the castle and tells Hamlet of his murder by his brother Claudius. This spurs Hamlet's long and labyrinthine quest to gather evidence against his uncle and enact his revenge. The play ends with a gargantuan body count, including the death of Hamlet himself, as well as nearly every other major character in the play. 
The themes of the play are existential and academic. Hamlet is a master of words, and nearly drives himself mad with his own indecisiveness ("To be or not to be, that is the question..."). His doubt, his confusion, and his emotion create a mountain of missteps. 
Hamlet is a complicated character. He is not exactly a hero, nor is he a villain. He is a tortured academic trying to do right by his family. And the nuances of this character are what have allowed endless reinterpretation and analysis.