The wife symbolizes common decency and faithfulness. There is not much said about the wife, beyond what she does: knowing that her husband loves animals, she gets him lots of pets; she apparently suffers quietly through her husband's drunken outbursts. At one point the narrator says that "he offered her personal violence," meaning he struck her. Even so, she stays with him even after they lose everything in a fire.
The wife serves as a contrast to the narrator's madness and paranoia. In a way, his perverseness is triggered by her patience. In fact, a central theme of the story is this notion of "perverseness," or the desire to do that which is forbidden or destructive. As the narrator falls into alcoholism, the wife comes to symbolize his old, compassionate, animal-loving self. As this contrast becomes more evident, it's clear that his loathing of the cat is a stand-in for his loathing of his wife, and both feelings are rooted in self-loathing.
It is the narrator's wife who points out that the white markings on the second cat are beginning to resemble something—a gallows. In a way, her perception is a kind of foreshadowing of her own death. The wife's death, in defense of the cat, is both a genuinely shocking twist and a kind of self-murder. When the narrator walls up the corpse of his dead wife, he is also walling up his own sanity.