Moral systems are interlocking sets of values, practices, institutions, and evolved psychological mechanisms that work together to suppress or regulate selfishness and make social life possible.
In my research I have found that there are two common ways that cultures suppress and regulate selfishness, two visions of what society is and how it ought to work. I'll call them the contractual approach and the beehive approach.
The contractual approach takes the individual as the fundamental unit of value. The fundamental problem of social life is that individuals often hurt each other, and so we create implicit social contracts and explicit laws to foster a fair, free, and safe society in which individuals can pursue their interests and develop themselves and their relationships as they choose.
Morality is about happiness and suffering (as Harris says, and as John Stuart Mill said before him), and so contractualists are endlessly trying to fine-tune laws, reinvent institutions, and extend new rights as circumstances change in order to maximize happiness and minimize suffering. To build a contractual morality, all you need are the two individualizing foundations: harm/care, and fairness/reciprocity. The other three foundations, and any religion that builds on them, run afoul of the prime directive: let people make their own choices, as long as they harm nobody else.
The beehive approach, in contrast, takes the group and its territory as fundamental sources of value. Individual bees are born and die by the thousands, but the hive lives for a long time, and each individual has a role to play in fostering its success.The two fundamental problems of social life are attacks from outside and subversion from within. Either one can lead to the death of the hive, so all must pull together, do their duty, and be willing to make sacrifices for the group. Bees don't have to learn how to behave in this way but human children do, and this is why cultural conservatives are so heavily focused on what happens in schools, families, and the media.