"A little more than kin and less than kind"
[Hamlet, 1.2.67] Hamlet says this in an aside. It means that Claudius is both his uncle and stepfather, but a bad person. Here, kind has three meanings: the modern sense meaning "nice", the original meaning of "ancestral family", and meaning that developed during Shakespeares lifetime of "natural" or "proper" in manner. Hamlet eludes that Claudius is not a nice person, not a direct blood relative, and involved in less-than proper or natural relationships (like with Gertrude).
"O, that this too, too sullied flesh would melt."
[Hamlet,1.2.133] Hamlet says this in his first soliloquy. This is the first time he speaks about suicide, wishing that his stained (defiled) skin would just fall off his bones and decompose. Hamlet is upset at his mother's marriage to Claudius, and not being allowed to return to Wittenberg for school. He's so upset with his mother, that he just wants to give up, but later in the soliloquy states that he can't commit suicide for religious reasons.
"...frailty, thy name is woman!"
[Hamlet, 1.2.150] Hamlet says this in his first soliloquy. This is said about his mother, and is both a misognyistic comment as well as one regarding his mothers incestousness. It essentially means that woman are weaker than men, but in context suggests this is meant on a moral grounds regarding relationships.
"Get thee to a nunnery. Why wouldst thou be a breeder of sinners?"
[Hamlet, 3.1. 131-32] Hamlet says this to Ophelia. Here, nunnery is an ironic double entendre, meaning both nunnery, and slang for brothel. He wants her to go to a nunnery so that she won't give bith to sinners (like himself, his mother, Claudius, or perhaps any child born from a *****).
"This above all: to thine own self be true."
[Polonius, 1.3.84] A remarkably brilliant quote from the dim-witted Polonius, spoken to Laertes before he leaves for France.
"In my mind's eye"
[Hamlet, 1.2.247] Hamlet speaks this phrase, coined by Shakespeare, he tells Horatio that he can see his dead father only in his head. This is before Hamlet has seen the Ghost, and in the part of the play where Horatio tells him he has seen his father as a Ghost.
"Something is wrotten in the state of Denmark."
[Marcellus, 1.4.100] Marcellus, a Danish soldier, says this line to Horatio, after Hamlet has forbid them to follow the Ghost with him. It lets the audience know that something's up, and sets the stage for the rest of the play to unfold (immediately after is the scene with Hamlet and the Ghost).
"Though this be madness, yet there is method in't."
[Polonius, 2.2.223-24] Polonius says this in an aside during a conversation with Hamlet. Immediately preceeding this line Hamlet is mocking Polonius by naming all of the physical attriubutes of "an old man" (Polonius), and by saying Polonius could grow as old as he if he could go backwards in time. Here, Polonius interprets what Hamlet is saying as nutty, but also recognizes a method to the madness (that Hamlet seems confident and composed in what he's saying even if it makes no sense to Polonius). The quote also brings up the issue of weather or not Hamlet's madness is feigned, perhaps a form of method acting.
"O villain, villain, smiling damnèd villain!" & "...one may smile and smile and be a villain."
[Hamlet, 1.5.113 & 115] Hamlet says these two lines in his soliloquy that immediately follows the Ghost's exit. The villain is Claudius, who Hamlet has just discovered murdered his father. The fact that Claudius is all smiles is an example of the theme of deception in the play, that Claudius not only decieved Hamlet, but all of Denmark.
"O, what a rogue and peasant slave am I!"
[Hamlet, 2.2.577] Hamlet says this in his soliloquy that ends Act 2 scene 2. At this point, he has discovered that Rosencrantz and Guildenstern are spying on him for Claudius. Hamlet has asked a group of traveling players (actors) to do a speech for him. Hamlet is shamed that he has displayed less passion in avenging his father's murder than the actor. In this soliloquy Hamlet bashes himself for not having the gall to do anything to Claudius yet.
"...brevity is the soul of wit"
[Polonius, 2.2.97] Polonius says this proverbial and ironic line in presenting his findings of Hamlet's madness to the King and Queen. The irony lies in Polonius' character who thinks himself a man of great cunning and wit, as well as in his words preceeding this. He gives a long winded introduction, full of superfluous words and contradictory rhetoric, but then says this perfectly brilliant statement, the contrast of which is very humorous.
"Suit the action to the word, the word to the action."
[Hamlet, 3.2.18-19] Hamlet says this to the actors, telling them that their lines have a reason for existing, so they should have a natural action and emotion as well. Their actions should correspond to the text and vice versa. This is an example of Shakespeare teaching the audience about acting, basically that they should keep it real and get into what they are doing to be believeable.
"The glass of fashion and the mold of form, th' observed of all observers."
[Ophelia, 3.1.167-68] Ophelia says this after Hamlet's "get thee to a nunnery" bit, and basically has just witnessed the man she loves acting crazy and telling her he doesn't love her. Here, she describes Hamlet (pre-madness) as having been the mirror (or model) of proper behavior, every bit the gentleman, and the model of attractiveness and self-disposition. (This Hamlet she describes is quite different from the one we see)