Hamlet Act III, Scene i Summary

In a room in the palace…
Claudius questions Rosencrantz and Guildenstern about whether they've gotten any closer to figuring out why Hamlet "puts on" this madness.
The two friends say that Hamlet admits he's been out of sorts, but he won't tell them why. If anything, they say, Hamlet's been pretty good at not saying much. Though he received them like a gentleman, his kindness seemed a little forced.
Rosencrantz reports that Hamlet was full of (nonsensical) answers to their questions, but doesn't ask many himself. Still, they say, Hamlet's excited about the players that are coming, and he asked Rosencrantz and Guildenstern to make sure the King and Queen come to the play.
Claudius is happy to attend, and especially pleased to hear that Hamlet's found some delight (though of course Claudius doesn't know this delight is based on the prospect of his being found guilty).
As planned, Claudius and Polonius set a trap for Hamlet: Ophelia is the bait. The two men will hide and wait to see what happens when Hamlet encounters Ophelia, hoping they'll be able to judge from Hamlet's interaction with the girl whether he's been driven mad by love, or by something else.
Gertrude accepts Claudius's plan, and says to Ophelia that she hopes it's Ophelia that has made Hamlet unlike himself. That way, Ophelia's virtues will bring Hamlet back around to his senses.
Ophelia, ever pliant, agrees with the Queen in hoping that this is the case.
Polonius then instructs Ophelia to walk around reading a prayer book, which will seem a reasonable excuse for her to be alone. Polonius quips that piety and devotion are often a good cover for wicked deeds.
Claudius says in an aside that Polonius's words are like a whip to his conscience. He notes he's covered over his wicked deed, but it doesn't make his deed any less ugly. Claudius declares he bears a heavy burden.
Hearing Hamlet approach, everybody clears out so Hamlet can privately deliver one of the greatest speeches of all time – to nobody but himself.
Hamlet ruminates on suicide, saying the question is "to be, or not to be."
He compares death to a little sleep. He claims death wouldn't be so bad, as it would end all the suffering of being alive. The only catch is that, in death's sleep, we don't quite know of what we'd dream. According to Hamlet, that consideration alone is enough to make us pause before killing ourselves.
Hamlet then lists off all of the things we'd escape by being dead, like being spurned in love, except that to escape those miseries by choosing death (of which we are ignorant, since no one comes back) is to hedge your bets. It seems better to put up with the bad things you know about in life than to run off into death and the unknown suffering that it may hold.
Hamlet concludes that most people choose life because death is a great unknown, like an "undiscovered country."
Hamlet then spots Ophelia reading her religious book, and closes his speech by saying he hopes the girl will count all of his sins (and ask for their forgiveness) in her prayers.
After a little small talk, Ophelia tries to return the letters and presents Hamlet gave her when they were in a relationship.
Hamlet insists that he never gave her anything, and asks if she's honest (playing on the meaning of "honest" as "chaste"). He suggests if she is honest and beautiful, then she shouldn't let the traits mix, as beauty corrupts honesty faster than honesty can influence the beautiful. (Basically, he's saying that pretty girls are easy to have sex with, and chaste girls aren't very attractive.)
Hamlet then tells Ophelia that he loved her once, though she shouldn't have believed it, because he claims that he actually didn't love her. (Their Facebook status is clearly "It's complicated.")
Hamlet then tells Ophelia to be off to a convent (the famous "get thee to a nunnery"), as it would be unwise for her to love him.
He then describes himself as fairly virtuous, though he could be rightfully accused of all sorts of awful things: Hamlet describes himself as "proud, revengeful, and ambitious." (Get your highlighter on that one – it's important.)
Hamlet further suggests that things would be worse if he had the thoughts, imagination, and time to make real all his darkest desires. Such men should have no place in the world, except that all men are such men. Thus, Ophelia had better get herself off to a convent, where it's safe.
Hamlet then asks Ophelia where her father, Polonius, is. She says he's in his own house, and Hamlet replies it's for the better that Polonius's foolishness is quarantined.
As a parting shot, he points out that even if Ophelia is a regular girl, marriage would still be awful. Actually, he says, if she has to marry, Ophelia should marry a fool, as wise men know that women only make men into "monsters" Note: That's a reference to the idea that all women turn men into "cuckolds" (men who are cheated on by their wives), who grow horns, like monsters.
After all, Hamlet says, women are monstrous anyway, painting their faces and frolicking about so men think they're cute, when really they're just going to ruin everybody's life. Women pretend to be dumb, he says, when they're really in it for the sex. Most importantly, he says it's this that's made him mad (which makes us wonder whether he knows he's being watched, and is putting on a great show).
Now that Hamlet has dismissed half the planet as faithless (because they're women), he suggests that there be no more marriage, ever. "Those that are married already—all but one—shall live, the rest shall keep as they are," he says.
Hamlet exits, and Ophelia essentially throws her hands up, exclaiming Hamlet has transformed for the worse. Though she had once "suck'd the honey of [Hamlet's] music vows," he now sounds out of tune.
Claudius and Polonius enter the scene. Claudius announces that actually, Hamlet isn't lovesick. Claudius notes that, though what Hamlet said could have been better constructed, it certainly didn't sound like the ravings of a lunatic.
The King realizes that this is a most convenient time to send Hamlet off to England because the change of scenery might do him good.
Polonius agrees that sending Hamlet off is a good idea, though he's still sure that the root of Hamlet's trouble is scorned love. Polonius recommends that Hamlet be left alone with his mom after the play, as she might convince him to reveal the true source of his grief. Polonius intends to hide and listen, while the mother and son confer. If Gertrude doesn't get any good information out of him, well, then it's off to England for the silent son.

Act III, Scene ii

Act II, Scene ii