Adulthood is generally something that is marked by a specific rite in mot cultures. To pass from childhood to adulthood, a person must accomplish a certain goal and become a "productive member of society."
Childhood is the first stage. Self-centeredness is the key thing to note here. Children are notorious for only caring about themselves. Some "adults" inflicted with this same desire.
Adolescence shows signs of change. This is a transition period, and some don't make it out. People here care about others, and try to care for themselves. For some reason or another, this just can't happen. Adolescents understand responsibility to an extent, but cannot implement this idea in their lives. Many cultures don't have this stage as something official. It can be considered as a "beginner adult."
In American culture, I believe this something that not everyone has to go through, as I hope to outline below. There are several different paths to adulthood, but the outcome is the same. Responsibility. Each path converges and the outcome is the ability to care for yourself and make responsible choices.
One way is taking care of yourself in all manners. In our culture, it is incredible how much time and resources it takes to raise a child. Some people still rely on their parents through the rest of their lives. For good or bad, these people are not truly "adults." They still rely on others to take care of them financially. For some, this can take well over the 18 years that makes us adults. For others, this happens at a much younger age, anywhere between 14-16.
Those who were raised in a middle-class background, know luxury, and they are accustomed to having it. In order to live that sort of lifestyle, it takes longer to "mature." Those who are from a poorer family tend not to have luxury and do not need it. Instead they can "mature" faster and become adult sooner as they do not require as much income to support themselves. The wealthiest of people typically inherit and don't reach adulthood; if they do the parent dies, and they are forced to run the business to continue their lifestyle, or they set out on there own. They don't usually have to become adults.
American life is very complicated. There are a few paths to becoming adults, and the one above I believe does happen the most. As an American, it can be possible to choose to not become an adult. Adulthood is mostly about mindset. You have to become responsible for your actions and doing so financially is way most do.
Having children opens the door to the choice of becoming an adult. For some it works, and for others it does not. Adult choose to care for the child and its well-being. Another "child" would not. Responsibility is key.
Disease and death are more tragic markers of adulthood, but they can transition a person from childhood to adulthood. Cancer and HIV/AIDS are two very powerful transition keys, but like other can be ignored (or not acted upon). The death of those closest to you can certainly trigger change, but not necessarily. These are marked by a choice to overcome and maintain responsibility for oneself.
This burden of responsibility does not negate fun. Adults can have fun as children can have fun. It may not take the same form, but it can be had nonetheless. Of course adult behavior tends to be more subdued and less rambunctious, but what must be understood is that adults choose to have fun in a manner that doesn't not cause harm or distress to others. Social gatherings are perhaps the most prevalent of adult fun.
Sometimes, adults do grow further and become overly responsible and forget to indulge in the self. Those who do so are denying themselves the pleasures that are rightfully theirs. Adults that do this can "regress," and it would be beneficial to do so.