In the United States you are considered a grown-up when you reach the age of 18. You are no longer legally connected with your parents, and you are entitled to the rights—and expected to fulfill the duties—of an adult American citizen. (You may vote and be called for military service, for instance.)
There is a good chance, though, that when you are 18 your body has not yet reached full maturity. Many people continue to grow for a few more years. Most are fully grown—at least in height—by the time they are 20 years old, though boys may keep on growing until they are 23.