I think being an adult is keeping an open mind, wanting to learn, being able to love and have compassion for others, to put others before yourself (as in the case of having children for example), to be able to not give in to peer pressure, to stand up for yourself and what you believe in. I could keep going...
Age doesn't always determine whether someone is an adult either, you could be 50 and still act/be like a spoiled child. Legally, 18 is the age of adulthood in this country, but adulthood is also determined by culture and customs. There are places where the age 16 is considered an adult. Some people/children are forced into adulthood, such as those that are in a poor family and must help take care of/are the sole provider of younger siblings/family members.
Adulthood is both a mental and physical state in my opinion.