The main conflict in the story is between the wife's wishes and her inability to realize them. Even though she doesn't explicitly say so, it's perfectly clear from her behavior that she's thoroughly bored with the daily grind of her family life and wants so much more than she's currently getting.

She's heartily sick of looking like a boy; she wants to pull her hair back in a big knot that she can feel. She also wants to have a nice cat sit on her lap and purr when she strokes her. In addition to such simple pleasures, the wife also wants something much more substantial: children of her own. She also wants her own house, which is associated in her mind with silver and candles.

The cat which she'd love to have and come sit on her lap is a symbol of refuge, something that would make her feel safe and secure. A cat, like a child, would give the wife something to care about, and it's clear that's the main thing that's missing from her life at present.

So long as she is unable to achieve what she wants in life, the wife's internal conflict will never be resolved, and she will continue to lead an unhappy, unsatisfying married life with her husband.