"The Woman" first appears in act one of the play. Willy is speaking with his wife, Linda, but a moment from his past (and specifically from his extramarital affair with "The Woman") intrudes upon his present. At first Willy hears the laughter of "The Woman" and tries to talk over...
"The Woman" first appears in act one of the play. Willy is speaking with his wife, Linda, but a moment from his past (and specifically from his extramarital affair with "The Woman") intrudes upon his present. At first Willy hears the laughter of "The Woman" and tries to talk over it, perhaps to prevent his wife from hearing it or perhaps in an effort to defeat and suppress the memory.
The moment when the memory overcomes the present coincides with the moment when "The Woman" says to Willy, "I picked you." Willy is so pleased to hear her say this. In fact, he is almost incredulous, asking her twice to affirm that it is true. The fact that "The Woman" chose him gives him a sense of validation that he is increasingly unable to achieve in his present reality. "The Woman" is thus significant because she helps us to understand, by way of contrast, what Willy lacks in his life in the present.
"The Woman" is also significant because she changes our emotional reaction to Willy. The discovery that he once had an extramarital affair perhaps reduces the sympathy which we might otherwise feel for Willy. Alternatively, Willy's affair with "The Woman" might actually increase our sympathy with him, if we interpret it as just another manifestation of his abject loneliness.
Another reason as to why "The Woman" might be significant is because we can infer that she—or rather, the affair that Willy had with her—is likely a cause of guilt and shame for Willy. This might help explain why he is so insecure, and it also might help explain, at least in part, his psychological breakdown.
The second appearance of "The Woman" is in act 2, when Biff discovers her and his father in a hotel room together. This moment is very significant because it explains so much about the fractious relationship between Willy and Biff. This is the moment when Biff stops idolizing his father and instead sees him as a liar. He yells at his father, "You fake! You phony little fake! You fake!"
In this moment, Biff loses his role model, and, at the same time, he loses his own identity. His own identity, after all, was made very much in the image of the man he thought he knew. "The Woman" is significant, therefore, not only because she helps us to understand Willy, but also because she helps us to understand Biff.