The preconventional moral stage, says Kohlberg, is based on the cognitive abilities of a person in Piaget's concrete operational stage. Moral decisions are egocentric (based on me) and concrete. So you can see how reward and punishment are the typical bases of reasoning in this stage. The conventional stage is based on the children's ability to "decenter" their moral universe and take the moral perspective of their parents and other important members of society into account. The postconventional stage is based on the adult's ability to base morality on the logic of principled decision making based on standards that are thought to be universalizable and not dependent on culture. Kohlberg's system was based on extensive research he and his students did with interviews in which they asked children and adults to give the reasons they had for moral decisions Kohlberg presented them with. So his stages and ages do not correspond exactly from Piaget, but you can see a tantalizing similarity.
Now we finally get to Gilligan. As a student of Kohlberg's, Gilligan was taken by the stage theory approach to understanding moral reasoning. But she disagreed with her mentor's assessment of the content of the moral system within which people developed. If you look at the table of Kohlberg's stages, you can see the question being answered in the third column is one of justice - the fourth stage gives this away with talk about duty and guilt. "What are the rules of the game?" seems to be the issue at hand. From her careful interviews with women making momentous decisions in their lives, Gilligan concluded that these women were thinking more about the caring thing to do rather than the thing the rules allowed. So she thought Kohlberg was all wet, at least with regard to women's development in moral thinking.
What set her off in thinking this was the fact that in some of Kohlberg's investigations, women turned out to score lower - less developed - than did men. Were women really moral midgets? Gilligan did not think so. In taking this stand, she was going against the current of a great deal of psychological opinion. Our friend Freud thought women's moral sense was stunted because they stayed attached to their mothers. Another great developmental theorist, Erik Erickson, thought the tasks of development were separation from mother and the family. If women did not succeed in this scale, then they were obviously deficient.
Gilligan's reply was to assert that women were not inferior in their personal or moral development, but that they were different. They developed in a way that focused on connections among people (rather than separation) and with an ethic of care for those people (rather than an ethic of justice). Gilligan lays out in this groundbreaking book this alternative theory.