Joseph Campbell, the mythologist made famous by his posthumous "Power of Myth" interviews with Bill Moyer in the late 80s, wrote many widely-read books about world mythology. A lot of academics complain that he sometimes treated individual myths and cultures too superficially, ignoring regional differences, beliefs, and local traditions to push his view that myths the world over share certain similarities and commonalities.
Campbell's "monomyth" thesis is very appealing and popular in an increasingly connected world where we're all struggling to cope with different cultures. It reassures us that we do have things in common with one another, that we will be able to understand one another, even though we speak different languages and pray to very different gods.
Monomyth controversy aside, Campbell frequently spoke of the "four functions of myth," which he analyzes and describes in several different books. (Off of the top of my head, I know he reviews them in Thou Art That.)
Campbell argues that mythology and religion -- "mythology is what we call other people's religion" he says -- serve important social and psychological functions for individuals and communities, and that's why these traditions of stories and beliefs get passed down.
Mythology cycles and individual myths help us come to grips with the world we live in, from the personal to the environmental scale. I may not be listing these in the same order he does, but in short, myths often serve one or more of four functions:
1. Myths explain or help us come to grips with the awful, inexplicable truths of life: that we have to consume and kill other animals (or plants) to survive, that we are born from somewhere, that we die, that we have sex. All these things are pretty hard to explain rationally, although science in the last few hundred years has made strides in explaining. But before that, and even now... those are some pretty serious and overwhelming bits of life experience, and there are a lot of myths about death, birth, killing and the reasons for consumption of other beings that make them a little less mysterious and scary.
2. Myths are a sort of proto-science answering "how did the world get the way it is?" or simply, "what IS the world?" Most cultures have myths about how the world was created, or at least how it's organized: elephants holding up the world on the back of a turtle swimming through space, or the separation of the heavens from the waters, or the cosmic egg, or what made the sun and moon, or why birds fly and animals hibernate. The Big Bang is a new myth, arrived at in a different way, but it still satisfies that psychological need to know and understand the cosmos/world around us.
3. Myths dramatize, teach, and give meaning to the patterns of human experience. This is slightly separate from function #1, which deals with biological realities, so to speak. There's also the patterns of life such as childhood, puberty, marriage, growing old, earning a living, etc. There's myths about the young man or woman setting out from home to find his/her place in the world. There's myths about the youngest or oldest child. There's myths dealing about the relationships between parents and children, or what it means to be an old woman, or an eligible bride, or a mother, or a father. Ideas of motherhood, old age, puberty, etc. vary by culture, and our thoughts and feelings about these roles and life passages are shaped by the myths we hear and tell about them.
4. Myths dramatize, teach, and reinforce the shared values and identity of a community, and instill a sense of pride or self-identity with that community. There are local heroes like Hercules, Brer Rabbit, Joan of Arc, George Washington and his cherry tree. There are myths like the Easter Bunny and the American Dream and the Final Frontier. Local myths, rituals, and seasonal festivals are very important in most religions and cultures. By participating in them, learning them, reenacting them, passing them on, people feel a sense of community (whether with their church or other local group) and at the same time are endorsing and teaching what's acceptable and "normal" in that group. Marriage between a man and a woman -- or polygamy. Being baptized. Not eating pork. Pledging allegiance to the flag. Exchanging presents stored under a tree.
Joseph Campbell's analysis of mythology shows a lot of common sense and builds on the more scholarly theories of academic mythographers. Also, his Four Functions of Mythology show how mythology is, in a way, the expression of the psychology and personality of a culture or community. Mythology and psychology are connected.