A conflict in any story is a battle between two opposing elements; the battle is usually fought between the protagonist and the antagonist. The protagonist in any story is the one who overcomes the conflict and changes as a result; the antagonist is the one who fights the protagonist, trying to keep him/her from succeeding. The resolution of any story occurs the moment the protagonist solves the problem that causes the conflict.
In Anna Sewell's Black Beauty, the title character is the protagonist, while the cruel men he encounters and the hard fates he suffers serve as the antagonists. Unlike most protagonists, Black Beauty has no control over the outcome of the story and no way to actually fight against his antagonists. His story is resolved the moment Fate decides to place him once again under the best care he has ever known.
As a colt, Beauty is warned by his mother of the evil in the world:
[T]here are a great many kinds of men; there are good thoughtful men like our master, that any horse may be proud to serve; and there are bad, cruel men, who never ought to have a horse or dog to call their own. (Ch. 3, Pt. 1)
She further advises him to always remain good, gentle, and hard-working no matter whom he winds up with. The conflict of Beauty's story begins the moment he begins seeing the truth of her words for himself. He is first put under the excellent care of Squire Gordon and his grooms, but under this care, he begins to witness the cruelty inflicted by other men upon other horses. As the story progresses, his conflict grows more intense as he begins experiencing the cruelty firsthand for himself under crueler and crueler masters, making him do harder labor.
The conflict begins to resolve when, driven to the point of collapse, he is ordered by a veterinarian to be given rest and proper care; he is then sold at a horse fair where he is luckily purchased by a kindly elderly man and his grandson, who nurse him back to health. The story reaches its resolution when he is sold one last time to some kindly ladies and winds up back under the care of one of the best grooms he has ever had--Joe Green:
My ladies have promised that I shall never be sold, and so I have nothing to fear; and here my story ends. My troubles are over, and I am at home; and often before I am quite awake, I fancy I am still in the orchard at Birtwick, standing with my old friends under the apple-trees. (Ch. 49, Pt. 4)
Though Beauty has not had an active role in the resolution, the story is resolved nonetheless because he is finally free of his antagonists. Plus, he has grown as a result of his experiences because he is now worldly-wise about the nature of people and the suffering in the world.