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 . . .

The Hecuba speech is significant in several different ways. It is significant to the plot, because it sparks Hamlet's idea of the play within the play. It is significant as a sign of Hamlet's psychology, because it shows how much he hangs on to the past and passions from the past, dwelling on them. This marks him throughout the play.
However, it is symbolically significant to the play because the queen in that play explodes with passion at the sight of her dead husband, so much so that the gods themselves would have cried if they'd seen it. This is a direct comment on how Hamlet's mother DIDN'T weep this way.

Greg

This soliloquy at the end of Act II scene ii shows Hamlet considering the way the acting he has just witnessed was able to access such powerful depths of emotion even though the performance was just "a fiction, ...a dream of passion".  In the small performance earlier in the scene, the actor describes the terror of Hecuba, a figure from Ancient Greece, who watches her husband Priam's murder at the hands of Pyrrhus.  In his soliloquy later on, Hamlet is impressed that the actor was able to express such empathy for Hecuba to the extent that the actor's "visage wann'd" and he had...

Tears in his eyes, distraction in's aspect, 
A broken voice, and his whole function suiting
With forms to his conceit...

Clearly the actor is so connected to Hecuba's emotion that his entire body communicates it fully even though Hecuba in reality means nothing to him: "What's Hecuba to him, or he to Hecuba, / That he should weep for her?"

Hamlet then moves on to attack himself for his inadequate expression of emotion in comparison to the actor's.  This is a familiar theme of the play.  Hamlet is frequently frustrated about his own inability to act decisively in response to the information that his father was murdered by his uncle Claudius.  Here, Hamlet's point is that he has much better grounds than the actor to express his emotions, but even so he is not able to.  He lets out a stream of abuse at himself, calling himself "pigeon-liver'd", a "John-a-dreams", and "A dull and muddy-mettled rascal", emphasising his over-thoughtfulness and lack of mettle (courage).

Finally, he moves on to concretise his plan to get the actors to perform a play about a murder similar to the one Hamlet suspects Claudius of committing.  Hamlet's theory is that Claudius will be so affected by the emotions of the play that he will reveal his guilt spontaneously.  He explains this in these lines:

                                         ...I have heard
That guilty creatures sitting at a play
Have by the very cunning of the scene
Been struck so to the soul that presently
They have proclaim'd their malefactions...

Overall, this speech reveals Hamlet's growing impetus towards action through the inspiration of the players.  It is also interesting to note that Shakespeare seems to be putting forth a case for the power of dramatic art to effect change in the world.

As the mother of Hector, Hecuba features in Homer's epic poem The Iliad, an Ancient Greek text.

This speech in Hamlet is spoken, of course, by Hamlet after the Player King has delivered his moving monologue about Hecuba.  Thisis an ancient tale, and, while it'sa tragic story, the characters and events of this story are nothing personal to the actor.  Still, the Player King is moved to tears as he tells the story.  Hamlet's speech which follows is generally known as the "rogue and peasant slave" soliloquy. In it, Hamlet has two key themes.  First, he berates himself for his comparative lack of emotion even for a just and personal cause and is amazed at the actor's ability to create such emotion for something totally disconnected from his life.  He says:

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?

Next, Hamlet notes that if this actor had half the reason, the "motive and cue for passion," that he did, he would be a rather wild man on stage, confounding all with his emotions and actions.  Yet, Hamlet says, he "can say nothing"--not even against a king who has usurped the throne by killing his own brother. He says he deserves all insults of word and deed for this lack of resolve and passion.  He continues his diatribe against himself: 

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!

Then comes the second portion of this soliloquy in which Hamlet makes a plan.  "About, my brain!" he says, and then he determines to reaffirm the King's guilt by enacting a play within a play (within a play, actually)--inserting a few lines into the play in order to catch the King off guard and ensure himself of Claudius's guilt.   

I'll have these players
Play something like the murder of my father
Before mine uncle: I'll observe his looks;
I'll tent him to the quick: if he but blench,
I know my course. 

In the end, Hamlet has--once again--made a plan to determine once and for all:

...the play 's the thing
Wherein I'll catch the conscience of the king.