Enter Polonius and Reynaldo:
The first part of this scene, between Polonius and Reynaldo, is not necessary to the plot of the play, and is often cut from performances. That's a shame, because it's lots of fun. Polonius thinks he is very wise, and Reynaldo, a sort of superior butler, knows otherwise.
Polonius is sending money and "notes" (maybe more advice) to his son, Laertes, but he's suspicious about what Laertes might be up to. He seems to assume that if Laertes is fooling around, he will lie about it, so Polonius has a plan. His "fetch of wit" (2.1.38) is for Reynaldo to find some friends of Laertes, and say certain things about him, such as "he's very wild," and see what the friends say. (Later, Hamlet, in trying to discover if the King is really guilty of murder, will use a similar device to discover the truth: he'll watch the King's reactions while "something like" the murder of King Hamlet is shown in a play.)
Polonius is quite proud of his plan, because, as he says, "Thus do we of wisdom and of reach, / With windlasses and with assays of bias, / By indirections find directions out" (2.1.61-63). Of course, when Polonius mentions "we of wisdom and of reach," he's thinking of himself, despite the fact that he takes too long to say all of this, forgets what he's saying, and contradicts himself. Reynaldo politely tolerates all of this, and goes about his business.
Exit Reynaldo. Enter Ophelia:
Just as Reynaldo leaves, Ophelia comes rushing in, badly frightened. Without warning, Hamlet has come into her closet (i.e., her study or sewing-room), seized her wrist, stared at her, sighed, and gone back out, all without saying a word. His clothes were unlaced and unbuttoned, and he had "a look so piteous in purport / As if he had been loosed out of hell / To speak of horrors" (2.1.79-81). It's interesting and puzzling that she should describe him very much as the Ghost might be described. However, Polonius isn't puzzled. He immediately jumps to the conclusion that "This is the very ecstasy of love" (2.1.99). He says that he is sorry he misjudged Hamlet, but he is most interested in rushing off to tell the King.
Polonius' exit lines, "This must be known; which, being kept close, might move / More grief to hide than hate to utter love" (2.1.115-116), are not the clearest that Shakespeare ever wrote, but they need to be considered carefully. He apparently means that if he doesn't tell the King that Hamlet is crazy because Ophelia dumped him, there will be more trouble than if he does tell. This implies or assumes a couple of things. First, the King is very interested in finding out what is wrong with Hamlet. And, since this is the first time we've seen any sign of the "antic disposition" that Hamlet said he might "put on," we may assume that the King hasn't seen any sign of it, either. Therefore, the "antic disposition" is probably not the reason for the King's interest in Hamlet's state of mind. We can guess that Claudius sees Hamlet as a potential political rival, and that Claudius senses danger in Hamlet's continued mourning for his father. Second, the phrase "hate to utter love" means that Claudius will hate to hear that the daughter of his close advisor has a relationship with Hamlet. This might lead us to guess that Polonius' real--though unstated--reason for putting a stop to the relationship was to make sure that he was on the right side: the King's.