I would say that to become an adult you would need to stop relying on your family for everything and you have people relying on you instead like a girlfriend/wife/boyfriend/husband/other? and/or a child.
There is also the point in which you've followed a particular career path and you're earning enough to sustain yourself without worry meaning you're independently paying for bills and sorting out life's problems.
Being 18 doesn't make you an adult, hell I know people in their 40's that still act like children and need to get a grip on reality which is funny because that's what they say to me. I believe once you are mentally mature enough to not ruin your life with a bunch of nights out and wasting your time instead of getting the right qualifications that's another way of becoming an adult.