Characterization is important because it galvanizes the plot forward, as do the characters' conflicts.
Characterization is defined as:
An author or poet's use of description, dialogue, dialect, and action to create in the reader an emotional or intellectual reaction to a character or to make the character more vivid and realistic.
Characterization generates plot and is revealed by actions, speech, thoughts, physical appearance, and the other characters’ thoughts or words about him.
In better knowing a character, the reader becomes more involved with the story. If the author is successful, the reader cares about the hero, and sometimes intensely dislikes the villain.
In Shakespeare's Hamlet, the audience experiences concern about young Hamlet. (The gravedigger seems to indicate that Hamlet is about thirty; the first Folio suggests that Hamlet is about sixteen, and has returned home from the university to attend his father's funeral. His age will make him a more sympathetic character.) The audience cares about the son who is disgusted by his mother's hasty remarriage to Old Hamlet's brother, and then discovers that his father was murdered by his own brother. Hamlet's father's ghost relays the information and calls upon Hamlet to avenge his death:
But know, thou noble youth,
The serpent that did sting thy father's life
Now wears his crown.
O my prophetic soul! My uncle! (I.v.43-46)
This introduces the primary conflict in the story: Hamlet must avenge his father's murder. However, during Elizabethan times, people believed that ghosts were not always "honest." They could appear in the guise of a loved one, and while they could not make a living person do something bad, they were thought to have the power to influence those still alive. If the ghost were evil, he might try to trick Hamlet into killing Claudius, which would be a mortal sin (regicide), and Hamlet would forfeit his mortal soul.
The second most important conflict, then, is for Hamlet to find proof that his uncle is actually guilty. For though Hamlet tells Horatio that the Ghost is honest, he needs to be certain. From this point, Hamlet will pretend to be insane in order to more easily look for the evidence that will allow him to act. Hamlet's indecision is frustrating for the audience, but he seems much more human for it. And we become more caught up in his distress.
During Hamlet's search, Claudius is trying to find out if Hamlet is a danger; while spies abound at the King's bequest, ultimately he tries to have Hamlet murdered—more than once. In this we gain a clearer picture of what Claudius will do to remain King.
Other characters also provide elements of characterization. In Act One, we learn from Laertes (Ophelia's brother) that Hamlet and Ophelia are sweethearts. Claudius reports later in the play that Hamlet is much loved by the people of Denmark, which makes killing him awkward.
We learn of the intelligence and depth of our hero, but also of our villains' capacity for evil. Laertes is weak and grieving so he is easily manipulated by Claudius to try to kill Hamlet. However, the characterization of Claudius provides the audience with the real villain—a character of great cunning and murderous intent.
Primarily, characterization gives us the thoughtful, honor-bound Prince, who flirts with suicide, berates his own "dull" purpose, and tries to fulfill his father's request.