The Awakening, which has long been hailed as a masterpiece of proto-feminist literature, has a lot to say about women and their roles.
The best example of who a woman "should" be is Adele Ratignolle. Edna calls her the 'mother-woman': she sacrifices herself for her husband and her children, which is seen as important for a woman to do. She does not exert herself physically, she has very little privacy or independence, and she cares far more about others than she does about herself. A woman, according to Adele (who typifies the societal view) should not ever consider herself. If she has desires, they are only OK if they agree with the desires of her husband. Adele stays at home during the day and accepts guests graciously.
Edna is not a 'mother-woman', though she is both a mother and a woman. In one memorable scene, Edna asserts that she would not sacrifice herself for her children. She loves her children and she often enjoys their company, but she does not see them as ultimately important. Sometimes Edna does not fulfill her social duties: on the day when she normally stays home, she decides to go out, which prevents her from receiving guests. This behavior makes her husband angry.
It is Edna's desire for independence and emancipation that really marks her as different from what a woman is supposed to be. She wants to live by herself, which is considered extremely inappropriate. She has a physical affair with Alcee and an emotional attachment to Robert, when a woman's role is to be faithful in all ways to her husband.
Mademoiselle Reisz is another example (perhaps a better one, even) of a woman who does not follow what a woman's role should be. She does not care what society thinks of her: she is blunt and often rude when a woman should be sweet, she is unmarried when a woman should only exist for her husband, and she is an artist on the piano when women should only practice art for how it can bring other people happiness. Mlle. Reisz helps Edna along the road to independence, since she is so independent herself.