The term adult describes any mature organism, but normally it refers to a human: one that is no longer a child and is now either a man or a woman. Coming of age is the event of becoming an adult, i.e. of entering adulthood.
Legally it means that one can engage in a contract. The same or a different minimum age may be applicable for e.g. parents losing parenting rights and duties regarding the person concerned, parents losing financial responsibility, marriage, voting, having a job, being a soldier, driving, travelling abroad, drinking alcoholic beverages (perhaps different ages for different categories), smoking, sex, being a prostitute (if legal at all), being a client of a prostitute, being a model or actor in pornography, etc.
The legal definition of entering adulthood varies between ages 16-21, depending on the region in question. Some cultures in Africa define adult at age 13, but most other civilizations consider this the teenager stage.