Digory is bold, curious, stubborn, and rash. He also has a healthy skepticism. These are the traits that drive the story. Digory also has some softer traits, such as love for his mother, sorrow that she is dying, and a sense of honor and respect for authority.
We see Digory's boldness and curiosity when he urges Polly to help him explore the attic and to use it to get down into the abandoned house. These traits, plus his rashness, come up again when the children get into the Wood Between the Worlds. Digory wants to explore the other worlds the pools lead to. Polly, who is more cautious, has to be convinced to do this. (Digory is stubborn during their many little fights.) Then, when Polly finally does agree, Digory almost charges off to another pool without marking the one that they needed to go back by. This would have left them lost in other worlds forever.
These same traits of Digory's come up again when the children discover the bell and hammer, with its warning rhyme, in the ruined palace in Charn. Digory is curious and wants to ring the bell. Polly tries to stop him, but being stubborn, bold, and rash, he wrestles with her and rings it anyway. This is what awakens the evil Empress Jadis, releasing her on London and eventually, on Narnia.
Digory's boldness and stubbornness come in handy later, when he must approach Jadis on her rearing horse in order to grab her ankle and put on the ring so as to get her out of London.
Digory is skeptical enough not to just believe anything that he is told. For example, when Uncle Andrew is telling Digory that as a magician, he (Uncle Andrew) is above the ordinary moral rules, adding, "Ours, my boy, is a high and lonely destiny," Digory realizes that Uncle Andrew simply means he should be able to do whatever he wants. He recognizes this same attitude later in Jadis. It is Digory's skepticism (and his loyalty to Polly) that later help him to resist Jadis when she tempts him in the hilltop garden.
These traits of being curious, stubborn, bold, and skeptical are exactly what make Digory such a good researcher and philosopher in later years, when he grows up to be "Professor Kirk."
Despite being somewhat rebellious, Digory has been well brought-up and he has a soft heart. Thus, he loves his mother and keeps trying to save her. He keeps his promise to Aslan, and though not impressed by just any adult (such as Uncle Andrew), he is able to submit to the authority and goodness of Aslan, recognizing that Aslan is worthy of his service. After meeting Aslan, Digory will carry the lion's influence with him all his life.