Taran is a young boy who is beginning to grow into a man. He lives in Caer Dalben with Coll, Hen Wen (an oracular pig), and Dalben. Dalben is a 379-year-old wizard with many mysterious powers. He owns an important book called the Book of Three. Sometimes, he reads the stories in the Book of Three to Taran. The stories are about important people, like kings, princes, and bards.
As a young orphan who doesn't know anything about his parentage, Taran is fascinated by these stories. He dreams that he could be unknowingly born of royal blood and that he too may have adventures one day. Taran leads a sheltered lifestyle in Caer Dalben, without much excitement or exposure to the outside world. The Book of Three symbolizes the excitement and glory he imagines will be waiting for him in the outside world. It tells stories of those who have done great deeds and acts of heroism. It is also something that is forbidden to Taran, who is not allowed to read the book himself.
When Taran sees Dalben napping, he decides to try and read the Book of Three without permission, thinking that the contents of the book will reveal to him new ideas and stories that Dalben is holding back from him. But as soon as he touches the book, it stings and burns his hands! Shortly after this, he leaves Cair Dalben alone to chase after Hen Wen after she escapes while under his watch. This catapults Taran into a series of adventures undertaken while he is trying to recapture Hen Wen.
The title The Book of Three symbolizes the adventures that Taran has throughout the course of the book, which were previously forbidden to him as too dangerous (just like the Book of Three was). It also symbolizes his need to learn more about the world and be a hero, just like the heroes in the stories in the Book of Three. He finds out that heroism is not quite all that it is portrayed as, and no longer values the excitement of violence and war the way he did when he first left Caer Dalben.
When Taran returns to Caer Dalben, he finds Dalben in his cottage, writing in the Book of Three. While this is not expressly stated, we can guess that Dalben is writing about Taran and his companions' adventures while Taran was away trying to find Hen Wen. This is a fitting culmination to the book because it shows how Taran's dreams were actualized through his own effort and bravery. In the end, he didn't merit glory because he found out his parents were royalty, or because of grand deeds. He earned it through his own bravery and effort, and by helping his friends succeed in their efforts to save Hen Wen and fight off the Horned King (one of the major antagonists in the story).
The title The Book of Three symbolizes Taran's intense need for adventure and glory, and his desire to learn and make his mark on the world.