What is the significance of Dumbledore's relationship with Harry?
Since Harry was a baby he has grown up without a father figure in his life. His uncle did not compare to any type of adult figure that Harry could look up to and constantly belittled and berated Harry. He refused to look out for Harry's best interests, only focusing on his biological son. As a result, Harry looked towards someone else to help fill the adult role, even more so a fatherly role. Dumbledore is kind and compassionate towards Harry, pushing him to become a better wizard than Harry thinks possible. Without the guidance and direction of Dumbledore in the first novel as well as the rest of the novels in the series, there are many times where Harry would have possibly died. With Dumbledore's guidance, Harry learned many things about himself as well as those around him. He was able to achieve great things through the support of Dumbledore and save the wizarding kingdom through learning from Dumbledore.
In the first novel of the series, Harry and Dumbledore occupy the classic archetypes of Hero and Mentor, respectively. Dumbledore is a respected older wizard who spouts endless wisdom and seems to be omniscient, keeping careful track of all the events in his domain. As to the significance of this relationship, the first point is that Dumbledore provides guidance in Harry's most difficult times. At the very of the book, Dumbledore helps Harry through the recent events (even though he does not reveal everything). Another, perhaps even more important point, is that Dumbledore sits back sometimes to allow Harry to grow for himself. Rather than taking his hand through every baby step, Dumbledore observes from the shadows to see how Harry develops on his own. A prime example of this is the part with the Invisibility Cloak/Mirror of Erised. Dumbledore gave Harry the cloak without putting his name on it and let Harry figure out the cloak's abilities on his own. Then, when Harry used the cloak to roam around the castle after hours, thus coming across the Mirror of Erised, Dumbledore turned the other way, though it is implied that Dumbledore knew this rule-breaking was going on. He only stepped in to warn Harry gently about the mirror's negatives and to inform the boy that the mirror would soon be changing locations. As such, Dumbledore's position as Harry's mentor walks a careful line between offering advice and letting him grow.
Dumbledore is Harry's mentor. He plays the role every aged wizard plays in a fantasy novel: bloody wise old fool who talks in endless riddles and could probably have prevented a whole bunch of deaths if he bothered to speak openly. But seriously, he teaches Harry the power of love, friendship, and loyalty. He offers consoling advice and trains him to defeat Voldemort. Helping Harry is Dumbledore's way of making up for his own past errors.
Dumbledore encompasses many qualities that Harry was forced to live without for eleven years. He provides Harry with a father figure. He is warm and nurturing- but, in keeping with a masculine character type- never coddles Harry. He pushes Harry to new and greater levels of acheivement throughout the novels. Specifically, in the first novel, he is one of the first adult characters (after Hagrid) that really listens to Harry. Like any good mentor, he listens and responds - usually without actually telling Harry what to do. He leads Harry to the right path without pushing him in any one given direction. That is Dumbledore's true magic- he always understood the grave importance of CHOICE.