As you probably all know, the first woman mentioned in the Bible is Eve who incidentally is not called Eve until after the fall. In the beginning, Eve is just called "the woman." In fact, as I flip through my copy of the New American Bible I am surprised to discover that the first time the man is called Adam is Genesis 4:25. What does Adam mean? Here is the link to an interesting article written by a Rabbi on the meaning of Adam's name: The Meaning of "Adam":Insights into the Hebrew Language.
God creates the woman for the man because "it is not good for the man to be alone" (2:18a). From the beginning, we see that God intends man and woman to be together, complementing each other. It is possible that God is not speaking here only about marriage although he certainly is speaking of that as well. It is possible that God is speaking of life in general. Modern science has proven that men and women think differently. In male dominated nations like China where women are aborted, there are great deficiencies. The same is true in male dominated corporations. Men by themselves are in a handicap. As I write this post on a woman, I find that it is impossible to speak solely on Eve. Humanity only begins when the man and the woman are together. Adam alone is lost. Eve is not just important because of the Fall (after all Adam ate the fruit as well) but because she is the "mother of all the living" (3:20). It is because of the fact that she is the mother of all the living that the Fall is so significant. Eve's choice affects the rest of the world, eternally. What a powerful role it is then to be a mother!