What are the theme and plot of Hamlet by William Shakespeare?

Shakespeare's Hamlet is a play so deeply embedded with a myriad of thematic ideas that perhaps it is best to begin with the plot. To shorten five acts down to a few paragraphs does not give full justice to the nuanced nature of this tragedy but here is the general gist:

A youngish Hamlet is home grieving the untimely death of his father when he is visited one late night by his father's ghost. The ghost, dressed for battle, intimates he was murdered by his brother, Claudius, who has subsequently taken the throne. The ghost goes on to express his frustration that his widow, Gertrude, has quickly cast aside his memory and married the murderous brother. With this knowledge and at the ghost's urging, Hamlet vows to take revenge on his uncle.

Grappling with the weight of what he must do, Hamlet proceeds through the next four acts struggling to follow through with his promise. He appears to decide that feigning madness will allow him to get close to Claudius, but first he must sort out his relationship with Ophelia. Because Hamlet seems repulsed by his mother's incestuous relationship, he pushes back against Ophelia's mild advances. Hamlet seems to understand, too, that Ophelia is being used as a pawn by her father Polonius and the King, as they strive to determine the nature of Hamlet's madness.

By Act IV, the nature of the tragedy comes to light. Hamlet kills Polonius, who was hiding behind a curtain, eavesdropping on Hamlet's conversation with his mother. Ophelia, deeply grieving over the loss of her father and Hamlet's affections, drowns in a river. Whether it was suicide or an accident we will never be certain, though dialogue from the grave diggers suggests the former. Hamlet has been sent to England by the king, who has instructed the English to kill his nephew-son. Hamlet concocts an elaborate ruse to avoid death and his former schoolmates, Rosencrantz and Gildenstern, are killed instead.

Laertes, Ophelia's brother, returns from France upon hearing the news of his father's and sister's deaths and comes prepared to take vengeance. He challenges Hamlet to a duel and aided by the king, attempts to kill Hamlet with a poison-tipped sword. King Claudius also has a poisoned cup of wine to use if necessary, but Queen Gertrude drinks from it accidentally and quickly dies. Hamlet, in a struggle with Laertes, switches swords with him and then strikes a blow, one that quickly turns lethal.

With nothing to stop him, and also mortally wounded from a blow from Laertes, Hamlet is able finally exact his revenge on Claudius, forcing the poisoned wine down his throat. At the play's conclusion, Hamlet begs his faithful friend Horatio to tell his story and he dies as Fortinbras, who was slowly storming the castle, enters ready to take the throne.

Thematically, we have madness and the nature of madness, gender roles, revenge, and mortality.