What It Is
The play within the play: The Murder of Gonzago, which Hamlet commissions from the actors and stages before Claudius, with the crime against his father reproduced on stage, and which he names, on the spot, The Mousetrap. It is the trap set to catch the king: if he betrays himself on seeing the poison poured into the ear of the sleeping king, his guilt is proven. The title is the character’s improvisation, and the motif of the rat extends, two scenes later, to the queen’s chamber, where the first creature caught is Polonius, hidden behind the curtain.
Where It Appears
In scene 2 of Act II, Hamlet asks the actors for the play The Murder of Gonzago and, alone, decides to test the king through the theater, after censuring himself for his own inaction.
The play’s the thing / Wherein I’ll catch the conscience of the King.(II.2)
In scene 2 of Act III, Hamlet asks Horatio to watch the king’s face; the pantomime and the play reproduce the murder in the orchard, and Claudius rises, orders the lights lit and leaves. Asked the name of the play, Hamlet answers The Mousetrap, and adds that it does not touch those who have a soul free of guilt.
The Mousetrap. Marry, how? Tropically. [...] Your majesty, and we that have free souls, it touches us not.(III.2)
In scene 3 of Act III, the proof the play tore from the king takes the audience: Claudius, alone, confesses the crime and tries to pray, and Hamlet, who finds him kneeling, spares him.
May one be pardon’d and retain th’offence?(III.3)
In scene 4 of Act III, the rat trap moves to the queen’s chamber: Hamlet thrusts his sword through the curtain behind which someone stirs, convinced that he is catching the king, and the rat is Polonius. In the same scene, Hamlet suspects that Rosencrantz and Guildenstern lead him into a trap on the journey to England.
Hamlet, on hearing someone behind the curtain in his mother’s chamber:
How now? A rat?(III.4)
In scene 2 of Act V, the motif closes the cycle: Laertes, wounded by his own poisoned blade, compares himself to a woodcock caught in its own snare and says he is executed for his treachery.