Wikipedia says:

Father of Ophelia and Laertes, and Lord Chamberlain to King Claudius, he is described as a windbag by some and a rambler of wisdom by others. It has also been suggested that he only acts like a "foolish prating knave" in order to keep his position and popularity safe and to keep anyone from discovering his plots for social advancement. It is important to note that throughout the play, Polonius is characterized as a typical Renaissance courtier, who pays much attention to appearances and ceremonious behaviour. Some adaptations show him conspiring with Claudius in the murder of King Hamlet, although Shakespeare himself gives no indication of this.

Polonius's most famous lines are found in Act 1, Scene 3, when he gives advice to his son Laertes, who is leaving for France, in the form of sententious maxims. He finishes by giving his son his blessing, and is apparently at ease with his son's departure. However, in Act 2, Scene 1, he orders his servant Reynaldo to travel to Paris and obtain information about whether Laertes is involved in vice there.

Laertes is not the only character that Polonius spies on. He is fearful that Hamlet's relationship with his daughter will hurt his reputation with the king and instructs Ophelia to "lock herself from [Hamlet's] resort." He later develops the belief that Ophelia's rejections of Hamlet's affections have caused the prince to lose his wits, and tells this to Gertrude and Claudius, claiming that his reason for commanding Ophelia to reject Hamlet was that Hamlet was above her station. He tests his theory with spying and interrogations.

In his last attempt to spy on Hamlet, Polonius hides himself behind an arras in Gertrude's room. Hamlet deals roughly with his mother, causing her to cry for help. Polonius repeats the request for help and is heard by Hamlet, who stabs through the arras and kills him.

The death of Polonius causes Claudius to fear for his life, Ophelia to go mad, and Laertes to seek his revenge.

Sparknotes writes: