Intro
So Hamlet's father was king until some guy named Claudius came along, killed him, married Hamlet's mother, and took over the throne. Hamlet is, by custom, supposed to avenge his father's murder by going ahead and killing Claudius himself. But, for whatever reason, he just doesn't do it.
The thing that critics have been puzzling over for ages is: why the heck can't Hamlet just get on with it and kill Claudius?
Let's look at the passage below. It's a soliloquy by Hamlet in which he's kicking himself for being such a procrastinator. Hamlet's just finished watching a play, and he compares himself to the actors ("players") in the scenes he's just finished watching.
Quote
Hamlet: Now I am alone.
O, what a rogue and peasant slave am I!
Is it not monstrous that this player here,
But in a fiction, in a dream of passion,
Could force his soul so to his own conceit
That from her working all his visage wann'd,
Tears in his eyes, distraction in's aspect,
A broken voice, and his whole function suiting
With forms to his conceit? and all for nothing!
For Hecuba!
What's Hecuba to him, or he to Hecuba,
That he should weep for her? What would he do,
Had he the motive and the cue for passion
That I have? He would drown the stage with tears
And cleave the general ear with horrid speech,
Make mad the guilty and appal the free,
Confound the ignorant, and amaze indeed
The very faculties of eyes and ears. Yet I,
A dull and muddy-mettled rascal, peak,
Like John-a-dreams, unpregnant of my cause,
And can say nothing; no, not for a king,
Upon whose property and most dear life
A damn'd defeat was made. Am I a coward?
Who calls me villain? breaks my pate across?
Plucks off my beard, and blows it in my face?
Tweaks me by the nose? gives me the lie i' the throat,
As deep as to the lungs? who does me this?
Ha!
'Swounds, I should take it: for it cannot be
But I am pigeon-liver'd and lack gall
To make oppression bitter, or ere this
I should have fatted all the region kites
With this slave's offal: bloody, bawdy villain!
Remorseless, treacherous, lecherous, kindless villain!
O, vengeance!
Why, what an ass am I! This is most brave,
That I, the son of a dear father murder'd,
Prompted to my revenge by heaven and hell,
Must, like a whore, unpack my heart with words,
And fall a-cursing, like a very drab,
A scullion!
Analysis
Sigmund Freud came up with a theory about Hamlet. He said that Hamlet's inability to kill Claudius is rooted in his own Oedipal complex. What's that, you ask? Well, according to Freud, all little boys want to kill their father and sleep with their mother. Of course, they can't really do this, so, voilà: Oedipus complex.
Freud said that Hamlet can't kill Claudius, because Claudius has done exactly what Hamlet has always wanted to do: he's killed Hamlet's father and married his mother. So Hamlet really identifies with Claudius on an unconscious level, and that's why he just can't bring himself to kill Claudius.
Psychoanalytic critics like Norman N. Holland and David Bleich would argue that Hamlet has such a hold on us because it reflects our own psychological complexes as readers. According to them, we can't get enough of Hamlet because we identify with Hamlet, in the same way that Hamlet identifies with Claudius. We too want to kill Dad and sleep with Mom, or vice-versa, and so we respond so deeply to Shakespeare's play. It reflects our own unconscious issues back to us.