There are two major kinds of conflict: internal and external. Internal conflict is the struggle that one goes through inside. External conflict includes a struggle with another person, with nature, with God (or the "supernatural"), or with society. Hamlet faces almost all of these conflicts.
The one that is perhaps most central to Shakespeare's Hamlet is Hamlet's struggle within himself about whether the ghost that appears to him is really his father or a spirit sent from the devil. Hamlet tells the others that see the ghost that it is "honest," but he spends the rest of the play looking for proof to support what the ghost has said: that Claudius killed Hamlet's father. This is an internal struggle for Hamlet because he knows that it is a mortal sin to kill a king. If Claudius is innocent and Hamlet kills him, Hamlet will forfeit his eternal soul. After a great deal of skullduggery and the pretense of insanity, Hamlet devises a plan to prove Claudius' guilt by adding a scene to a play very similar to the murder of Old Hamlet:
…the play's the thing
Wherein I'll catch the conscience of the King. (II.ii.604-605)
Hamlet also experiences the conflict of man vs. man. Treachery in Denmark's royal court abounds, found in the snooping of Polonius; Ophelia's forced cooperation by the King and her father to spy on Hamlet; the willingness of Rosencrantz and Guildenstern to betray Hamlet in order to ingratiate themselves with the King; and, Laertes burning desire to kill Hamlet when the prince mistakenly kills Polonius.
Man vs. the supernatural appears in several instances: is the Ghost real or an apparition sent by the devil? Hamlet also bemoans the fact that God has declared that suicide a sin:
O, that this too too solid flesh would melt,
Thaw and resolve itself into a dew!
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter! O God! God! (I.ii.132-135)
We might argue that Hamlet faces conflict with society: his uncle is chosen to be King over Hamlet—though Hamlet seems more concerned about his uncle's embarrassing carousing and his mother's "o'erhasty" remarriage to his uncle.
The worst example of man vs. man that Hamlet experiences is Claudius' repeated attempts to have Hamlet killed. First he sends Hamlet to England with Rosencrantz and Guildenstern who hold letters that call for the King of England's aid in executing Hamlet. The King addresses the English monarch:
...thou mayst not coldly set
Our sovereign process, which imports at full,
By letters congruing to that effect,
The present death of Hamlet. Do it, England... (IV.iii.66-69)
Claudius also convinces Laertes (devastated first by his father's murder, and later by Ophelia's loss—which he also blames on Hamlet), to take part in a plot to murder Hamlet. Forgetting his honor, Laertes allows himself to be manipulated into engaging in a "friendly" duel in which Laertes will poison Hamlet (who unknowingly poisons Laertes as well). Claudius also tries to get Hamlet to drink poisoned wine, which Gertrude mistakenly drinks. In trying to hold onto the crown for which he murdered his brother, Claudius succeeds in poisoning Hamlet (and Gertrude), but Hamlet turns the King's treachery back upon him before Hamlet succumbs to his injury at Laertes' hand.
Hamlet's promise to avenge his father's death, obtaining justice for his murder, ends in Hamlet's death—the result of a multitude of conflicts in the play.