the purpose of playing, whose end, both at the first and now, was and is to hold, as ’twere, the mirror up to nature

This line, which Hamlet delivers to the Players in Act 3, Scene 2 evokes the play’s examination on the blurred lines between performance and reality, as is discussed in the explanation for it in Quotes by Theme: Performance versus Reality (the fourth quote). However, its larger importance and reason why it is quoted so often to this day is because it represents William Shakespeare, the Bard, delivering to us a universal truth about theater from the distant past which remains true to this day.

HAMLET: Lady, shall I lie in your lap?
OPHELIA: No, my lord….
HAMLET: Do you think I meant country matters?
OPHELIA: I think nothing, my lord.
HAMLET: That’s a fair thought to lie between maids' legs.
 

Hamlet and Ophelia have this uncomfortable exchange in Act 3, Scene 2, shortly after their disturbing encounter in the prior scene in which Hamlet attacked Ophelia’s character and virtue. Here, the level of discomfort remains high as Hamlet voices a series of inappropriate sexually innuendos to the young woman he had previously loved. When he asks, “Do you think I meant country matters?” he is referencing sex. (As general rule, if a word in Shakespeare sounds like it might have a sexual connotation, there’s a good chance that it does.) This quote is also discussed in Quotes by Character: Hamlet (the eighth quote) and Quotes by Motif: Misogyny (the fourth quote).

The lady protests too much, methinks.

Gertrude delivers this, her most famous line, as a critique of a character in the play that Hamlet stages in Act 3, Scene 2. By saying that the character (who she does not realized is based on herself) “protests too much,” she is essentially saying that she is insincere. You can read more about this quote in Quotes by Character: Gertrude (the third quote).

… do you think I am easier to be played on than a pipe? Call me what instrument you will, though you can fret me, yet you cannot play upon me.

Hamlet angrily says this at Guildenstern at the end of a tense encounter in Act 3, Scene 2 with his one-time friends Rosencrantz and Guildenstern as they try to trick him into giving them information that they can pass along to Claudius. Incensed by their disloyalty, Hamlet picks up a recorder (a type of flute) and asks Guildenstern to start playing it. When Guildenstern replies that he doesn't know how, Hamlet asks him if he thinks that he (Hamlet) is easier to manipulate than a pipe. He then tells Guildenstern that while he may be able to push his buttons, he cannot play him like a fool.

This uncomfortable exchange touches on some ideas that run throughout the play, including the unfaithfulness of Hamlet’s former friends (with the notable exception of Horatio) and the fact that while he may pretend to be mad, Hamlet is clearly still in possession of his mental faculties and his keen wit.