What is the main conflict in The Magician's Nephew?
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.
This is an interesting question, because actually I would want to argue that there are several conflicts in this great novel. However, if we are trying to identify the "main" conflict, I would say that it would have to be the conflict between Digory and Polly and Digory's uncle, Uncle Andrew. Of course, it is Uncle Andrew himself that uses the children as guinea pigs, sending Polly against her will into another world and then forcing Digory to go after her so she can return. Note how Uncle Andrew is presented in Chapter Two:
"I hope," said Uncle Andrew presently in a very high and mighty voice, just as if he were a perfect Uncle who had given one a handsome tip and some good advice, "I hope, Digory, you are not given to showint the white feather. I should be very sorry to think that anyone of our family had not enough honour and chivalry to go to the aid of--er--a lady in distress."
Uncle Andrew is therefore presented as a manipulative man who uses children to do what he is not brave enough to do himself, and blatantly blackmails Digory into fulfilling his commands without any moral compunctions. It is this conflict between the children and Uncle Andrew that starts the story and leads them on into the other conflicts that they face.