I will provide a brief summary of the story here, but for further study, please check out the comprehensive summary that Enotes has provided for The Red Pony at the link referenced below.
In the first section of the book, entitled "The Gift", the central character, Jody Tiflin, is introduced. Jody is a quiet boy who lives on a ranch in California. He has no playmates other than the dogs on the property, and occupies his time doing chores and playing alone. One day, his father, Carl Tiflin, brings him a pony which Jody names Gabilan. Jody cares for Gabilan and teaches him to wear a bridle, but the horse gets sick and dies, leaving Jody devastated.
In the second section, "The Great Mountains", an old man named Gitano comes down from the mountain range surrounding the ranch. He says he used to live on the land and has come home to die. Carl Tiflin tells the old man that there is no work and that they can't afford to keep him, but that he can stay for dinner and sleep one night in the barn. Jody takes Gitano to the barn and shows him Carl's horses, and the old man notices a horse named Easter, who is thirty years old and no longer useful. Carl makes a cruel comment about the old needing to be put out of their misery. In the morning, both Gitano and Easter are gone.
The third section is called "The Promise". Carl gives Jody another chance to have a pony. He arranges for one of his mares to be mated, but when the mare finally becomes pregnant and is ready to give birth, the colt is positioned wrongly inside her, and Billy, the hired hand, must kill her in order to save the colt. Jody is happy to have a horse again at last, but he is traumatized by the vision of the dead mare and Billy covered in blood.
The last section of the book is called "The Leader of the People". Jody's maternal grandfather comes to visit, and Carl is annoyed because the old man incessantly tells stories of the past, about "Indians and crossing the plains". In the morning, before Grandfather comes down to breakfast, Carl is complaining about him, ridiculing his stories, and sadly, the old man arrives and overhears every word. Grandfather is hurt, and later tells Jody he may leave early, saying he tells the stories because he wants others to feel the exhilaration he once did, and questioning the value of his past accomplishments. He laments that the West is gone, and that there is no place left to go.