A grown-up is a fully grown adult person. Grown-up as a noun is mostly used by children. As an adjective, grown-up refers to a mature style or taste.
Betsy dislikes when grown-ups ask her about school.
The children know that they should find a grown-up if they see a suspicious person.
The teacher thought the twelve-year-old girl's lipstick was too grown-up.
Grown-up as a noun is mostly used by children or when speaking with children. It is also used by adults to convey humor or sarcasm; otherwise they usually just say the noun "adult." English speakers use the related phrases "grown woman" and "grown man" to emphasize the fact that someone is an adult. For example: "Mother, please don't tell me what to do. I'm a grown woman and can make my own choices!"
Grown-up is related to the verb phrase “grow up,” which means “to reach maturity or become fully grown.” Don't forget that the adjective and noun have a hyphen (grown-up) but the verb doesn't (grow up).
As an adjective, grown-up was first used between 1625 and 1635, though the verbal phrase "grow up" already existed, in some form or another, since before the year 900. As a noun, grown-up has been used only since around 1805-1815. From Middle English growen, Old English grōwan.