At the beginning of the novel, the main conflict is the conflict between Harry Potter and the family with which he lives. Mr. and Mrs. Dursley treat Harry like an animal or a slave. He is forced to live in a small cupboard under the stairs, and he is given just enough food to keep him from starving. He is also bullied relentlessly by Mr. and Mrs. Dursley's large son, Dudley.
Later in the story, when Harry arrives at Hogwarts, the main conflicts are the conflicts between Harry and Professor Snape on the one hand, and Harry and another schoolboy, Draco Malfoy, on the other. Professor Snape seems to have an irrational hatred of Harry and even bullies Harry in his lessons. Malfoy, meanwhile, taunts Harry about his dead parents, telling him at one point,
I'd be careful if I were you Potter ... Unless you're a bit politer you'll go the same way as you parents. They didn't know what was good for them either.
While all of these conflicts are important, the most important conflict in this book, and throughout all of the subsequent books, is the conflict between Harry Potter and Lord Voldemort. Early in the story we learn that Lord Voldemort tried to kill Harry when Harry was just a baby. We also learn that in trying to kill Harry, Lord Voldemort killed Harry's parents. Harry and Voldemort meet at the end of the book. Lord Voldemort, having possessed the body of Professor Quirrell, tries to kill Harry again. Harry manages to escape, but Voldemort lives on, and the conflict is resumed in the subsequent books.
The conflict between Harry Potter and Lord Voldemort is the main conflict in the book because it epitomizes the conflict between good and evil. It is also the main conflict because it is the most consequential. Harry wants to defeat Lord Voldemort to get revenge for his parents' deaths, but he also needs to defeat Lord Voldemort in order to save the whole world. Lord Voldemort, on the other hand, needs to defeat Harry Potter in order to eliminate the only real threat to his power and to thus consolidate that power.