The antagonist in fiction is that character who has adopted a fixed solution to an inner vacuum or emptiness or threat.
This fixed solution is programmed to make this particular character right at all times, to rob power from others, and to dominate the world around them at the expense of everyone else. This usually takes the form of a weapon or power source of some kind, often one which saps the will or life energy of others.
This ‘solution’ - whether it be obvious, like Sauron’s One Ring, the Emperor’s Death Star, Voldemort’s Deathly Hallows, the White Witch’s stone knife, or less obvious like Bob Ewell’s fixed racist attitude in To Kill a Mockingbird, or the arrogance and pride of Lady Catherine de Bergh in Pride and Prejudice -- is also the thing that the protagonist ends up exploiting or turning to his or her advantage at the climax of the tale.
In stories that we call ‘realistic’ or Ironic, this fixed solution becomes more psychological, an immovable attitude or prejudice; as the tale edges toward the fantastic or purely Epic, the solution becomes less and less metaphorical and more and more actual and physical, an external object.
Of course, in no case is this thing a solution at all: in actuality it exposes the ultimate weakness of the character and his or her complete dependence upon a particular mechanism or idea. Once that idea is fundamentally questioned, everything crumbles for this figure.
In the rare cases that we know anything about this character’s past - usually in Tragedies or Ironies, like Macbeth or Claudius in Hamlet - we find that the idea or mechanism was created or built in a moment of extremity. It became a solution to that time of stress or threat and grew as time went on, becoming both more potent and more burdensome. Normally, though, in Epics and Comedies, we are not permitted a glimpse into the antagonist’s mind and all we see in the story is the mechanism being deployed.
That’s because the story is about the other character, the protagonist.
The protagonist could be said to be the character who, in a moment of extremity, doesn’t adopt a fixed solution but acts to resolve the extremity in some other way. Instead of, like the antagonist, seeking the safety and apparent security of a particular mechanism or idea, and then retreating into it and pumping more and more power into it until it forms a kind of calcified shell around him or her, the protagonist refuses to withdraw. This produces suffering, but also ultimately, victory.
The antagonist is not a rational creature: he or she ceases to live, ceases to examine life as it is, but adopts and hides behind a shield which becomes a weapon and a liability. It’s a substitute for living. Worse than that, as time goes on and real life grows and changes around the antagonist, he or she must pour more and more power into the mechanism to assert its rightness and authority.
Antagonists became trapped by their own most powerful idea.
Undermining that idea, destroying that mechanism, pulling that fixed solution apart, that is the task of the protagonist.
How he or she does that forms the story.
For more about antagonists and protagonists -- and much else besides -- please see my book How Stories Really Work.