The major conflict is Ahab's quest to destroy the whale, Moby Dick. This could be considered as man vs. nature or man vs. the unknown. Moby Dick represents evil incarnate to Ahab, the devil. Ahab is obsessed with killing the whale as its punishment for taking his leg. He's willing to risk his life and the lives of his crew to get his revenge. Starbuck, the first mate, tries to discourage Ahab from this suicidal voyage several times, but Ahab cannot be deterred from getting his vengeance.
Some critics disagree regarding which characters are the protagonist and the antagonist. Most see Ahab as the protagonist, and Moby Dick as the non-human antagonist. I tend to agree with this view, as Moby Dick, the symbol of evil in the world to Ahab, is the force that Ahab is fighting against. Other critics say that Starbuck is Ahab's antagonist because he attempts to dissuade Ahab from continuing the voyage to kill Moby Dick.
For a more detailed explanation of this novel and its characters, go to the link below.