In The Magician's Nephew, it's helpful to think of the conflict in terms of the external conflict and the internal conflict. Both types of conflict are important to the plot and characters in the book. External conflict refers to a force outside the main character that keeps him or her from achieving a goal or solving a problem. Internal conflict refers to forces within the character himself or herself that stand in the character's way of getting what he or she wants.
The external conflict for Digory, the main character, begins with Uncle Andrew. When Digory and Polly accidentally stumble into Uncle Andrew's study, he locks them in and experiments on them with his magic rings. This results in Digory and Polly eventually finding themselves in Charn, where they awaken Queen Jadis. From then on, the primary external conflict comes from Jadis, who follows them back to our world and then to the newly created world of Narnia. Jadis puts the characters in physical danger, and she also presents Digory with the temptation to disobey Aslan and steal an apple for himself.
Digory also faces internal conflict. At various points in the novel, Digory struggles with himself to make the right decisions and do the right things. At the beginning of the story, he has been crying and has (understandably) allowed his emotions to run away with him because of his mother's terminal illness. Learning to restrain his emotions and act maturely remains a conflict for him. For example, he must decide whether to follow Polly to wherever the magical ring has taken her, although that means giving in to the manipulation of Uncle Andrew. In Charn, he finds it impossible to resist ringing the bell, which wakes up the evil queen. Finally, he faces a great temptation to disobey Aslan and steal an apple for himself. He conquers that temptation relatively easily, but when Jadis suggests he steal an apple for his mother, he must decide not between good and bad but between good and better. This is a sophisticated test that requires maturity, and Digory passes it, choosing to obey Aslan and not listen to the evil queen. Digory shows even greater maturity by confessing his near lapse to Aslan, and Aslan praises him and rewards him with the apple that heals his mother.
The main external conflict in the novel is man vs. man in the form of Digory vs. Jadis. The main internal conflict is Digory vs. his immature desires. Digory's triumph over his immaturity allows him to win his external conflict with the evil Queen Jadis.