Polonius is the lord chamberlain, the chief counselor of King Claudius and father of Laertes and Ophelia. As he bids farewell to his son, who is leaving for France, he gives him a list of precepts that ends with "above all, to thine own self be true."
He controls his children. He forbids Ophelia to speak with Hamlet ("See to it, I command you.") and sends his servant Reynaldo to spy on Laertes in Paris, explaining the method: "See you now, your bait of falsehood takes this carp of truth."
He concludes that Hamlet has gone mad for love of Ophelia, and hides with the king to spy on the meeting of the two, in Act III, scene 1. In scene 3, he tells the king that he will hide behind the arras to hear the talk between Hamlet and his mother: "I will hide behind the arras to hear the talk."
He dies in the queen's chamber: when he is heard stirring and crying for help, Hamlet thrusts his sword through the curtain and says "How now? A rat?", convinced he has killed the king. Only afterward does he see that it is Polonius. Claudius admits in Act IV, scene 5, that they buried him in secret, and that this left the people restless; Laertes returns from France demanding explanations.