So where did the whole idea of stages come from? Switzerland and Jean Piaget. Piaget was a careful observer of children's thought and behavior in a wide variety of circumstances. His work is so vast that any claim to describe it in a few pages must fail. So I won't. I will instead talk about Piaget-for-us, which is fine, since American psychologists have regularly borrowed from Piaget what they liked and left much behind. For us, Piaget's central claim was that increases in reasoning skill over time were punctuated by shifts in perspective that could only be called qualitative change from one stage (or "type," if you will) of thinking to another. This is a thoroughly cognitive theory; Piaget could ignore the behaviorists from his mountain fastness in Switzerland. For Piaget, children start out as concrete and egocentric thinkers (infants even have to learn that objects persist when they are out of sight). As they gain more cognitive ability with age, they begin to be able to "decenter" and see things from another perspective. But they are still concrete in their approach to things. More experience and (here is a key) some cognitive reorganization eventually allow most people to become abstract thinkers.
So what does this have to do with Carol Gilligan and women? We are getting there. Gilligan was a colleague of Lawrence Kohlberg at Harvard. Kohlberg had applied Piaget's theory to the development of moral thinking. Borrowing from Piaget's "preoperational/concrete/formal" distinctions Kohlberg came up with the stage theory you see here