You're biologically an adult when you're done with puberty and stop growing. That stopping point varies from person to person, of course. I've stopped growing (as in I've been hoping for a grow spurt since seventh grade and it just never happened) and I'm not even sure I went through puberty (as in I never had any wet dreams and (thankfully) never had the ability to grow sideburns).
You're legally an adult at 18. "Adult" as in you can go to college and mooch off your parents and you can sign things with your own pretty signature which you've been practicing since third grade. Although I don't think 18 counts, because my definition of an adult includes alcohol, which doesn't happen until you turn 21. Adults go to bars and make "grown up smoothies" that you think are gross but can't wait to chug when you're old enough. Although I don't think 21 really counts either, because you can't rent a car until you're 25. Adults have the ability to rent a car they normally wouldn't be able to afford just to drive around for a few days to look cool.
You're sexually an adult when you choose loose your virginity. I gave my V card to a cute, musically inclined black-a-rican (i.e. black/Puerto Rican) when I was 17. Some people wait until they're thirty and married, while others (TROLLOPS) start in middle school and become teen parents and end up on MTV.
You're socially an adult when you can interact with other human begins with an open and understanding mind. This means you can talk to people who are any combination of the following without giggling or scowling or any of that bullshit: stupid, obnoxiously pretty, racist, 3-feet tall, covered in snot, gay, religious, homeless, wheelchair-bound, etc. Many people never reach this point. And even if they do, there's always something (i.e. a stupid obnoxiously pretty racist 3-feet tall gay wheelchair-bound jesus freak covered in snot) that makes them fall on the ground laughing. Or crying. Or both.
You're financially an adult when you stop asking your parents for money every day. You can pay your rent, your utilities, your cell phone, your internet and cable, your porn subscriptions, your clothes, your shoes, your food, your take out, your bar tab, your gas, your car insurance, etc. A lot of people get there. I'm mostly there (minus the car insurance my dad pays and the "Of course we'll help you out" when I call crying because my dog needs to go to the vet and I have $7.83 in my checking account).
All of these explanations of adulthood aside, I can finally tell myself that I am grown up. Why? Because today, just now, I powdered and vacuumed my carpets on my own free will. Not just vacuumed. No. That's something kids do when they're parents bribe them with ice-cream and beef jerky. I cleared the room of furniture. I dusting the carpet with a thin layer of Glade Carpet Cleaner and Refresher with a Pet Oder Booster (satisfaction guaranteed). I folded two loads of laundry (also very adult of me if I do say so myself) and waited for the powder to sink in. Then, in a fast, loud, white cloud of dust, I vacuumed. And re-vacuumed. And re-re-vacuumed. And re-re-re-vacuumed.
I did all of this by choice.
And now I choose to eat a very adult bean burrito.