In The Awakening, what is the "outward life which conforms, and the inward life which questions"?
The "outward life" describes the social persona—the mask individuals assume when out in the world among other people. This persona is taken on so an individual can comfortably engage in society without seeming like a freak or an outsider. The "inward life" is the person behind the social mask. This inner self may or may not agree with the rules that come with engaging in the public social world, hence the verb "questions."
In The Awakening, the protagonist, Edna, is someone whose outer and inner identities are at odds. While she appears to be a dutiful wife and mother, in truth she gains no satisfaction from either of these roles, even though society expects her to do so. She does not love her husband, even if she is fond of him in a casual way. She is not mad with devotion toward her two children either. At one point, the narrator mentions that, when the boys were staying with their grandmother for a long period, Edna rarely missed them:
Their absence was a sort of relief, though she did not admit this, even to herself. It seemed to free her of a responsibility which she had blindly assumed and for which Fate had not fitted her.
So while Edna does not hate her husband or children, she is only a wife and mother because society expects her to be, not because she wants to be. Realizing her unhappiness, Edna begins to question how worthwhile it is to go about the charade of performing as her "outer" mask.
Inside, Edna has the soul of an artist. She appreciates beauty and delights in sensuality. She loves to paint. She loves to swim in the ocean. She enjoys living alone. Over the course of the novel, Edna "awakens" to her true self, the person she wants to be. Unfortunately, her inner self is at odds with what society expects and, eventually, the loneliness of being an outsider drives Edna to suicide.
The distinction is pretty straightforward. There is an outer life, or a public face, that we wear for others to see, but at the same time there is a separate inward existence, one that questions (or rebels against) the social conventions that determine our outward life. There are many ways this split can be explained; at bottom, however, it comes down to the person you know yourself to be and the person you pretend to be for the benefit of others.
Edna's outer life, the life of being a wife and mother, is shown in Chopin's novel to be a lie. The thing that society expects Edna to be is not the thing she wants to be. Her love for Robert, and her affair with Arobin, are expressions of that realization but also examples of how dangerous this inward life can be. Her suicide can be understood as a reaction to the unbearable confinement of society, but I also wonder if the brief encounters she has with these other men are a factor. It’s not clear that Edna really understands what the inward life is demanding; she only knows that something is missing, and she kills herself out of despair at never being able to find it.
In Kate Chopin's The Awakening, the "outward life" mentioned in this passage refers to the moralistic set of narrowed expectations that phallocentric society imposes upon women, whereas the "inward life which questions" addresses the protagonist Edna's resolve against these restrictive standards. Initially, Edna leads a normative feminine experience, fulfilling her expected roles as a wife and mother. While she participates in these roles, she yearns for freedom and independence from these reductive norms.
As the novel progresses, Edna revolts against the patriarchal conventions of the time. Her actions start out subtly when she refuses to participate as a host, and escalate to her moving out of her husband's house and engaging in a physical affair. She eschews her prescribed gender duties, disregarding her maternal obligations and instead choosing to lead an independent life. Her refusal to ascribe to the gender norms of the time ostracizes her, and Chopin ends the novel on a dark note by having Edna commit suicide. This shows the oppression that women faced at the time as the detrimental force it truly was.
In The Awakening, the protagonist Edna Pontellier leads a life that does not make her entirely happy. She is married and has children which conforms to the standard social conventions of Edna's time. However, she inwardly questions whether or not she should try to break free from this life to find her own independence and happiness. While she goes through the motions in her everyday life, Edna continually questions whether or not it is right for society to force her (and other women) into such a narrow lifestyle. Soon Edna begins to neglect her children and she has a love affair with another man, proving that her inner questioning gets the better of her and motivates to seek a new lifestyle.