Adult seems hard to define in anything but biological terms, but on a more philosophical level, there's two moments when I've felt like an adult:
1. When I was able to see my parents as flawed. As a child I think we all have a built-in degree of admiration for most adults, but it's only in adulthood that you can see, sympathize with, and appreciate the flaws your parents have. You also work on not having or succumbing to these flaws.
2. When you realize that everyone is pretending and no one really has it as together as they like to appear. We're all just acting like adults, but all the props we have in life can be taken away.
3. It hasn't happened yet, but when my parents pass away I'll feel like an adult. Those are the only two people that have loved me in the way a parent loves a child and their no longer being around will have me thinking "this is it, I can never be a child again".