Modest and reserved in manner or behavior (typically used about women though there are no strict rule that it shall be only used that way)
A demure person can be described as polite and a little shy. A demure outfit is a modest one.
It may have derived from the Anglo-French verb demorer or demourer meaning to linger.
She was demure and reserved, a nice modest girl whom any young man would be proud to take home to his mother.
He was mesmerized by her demure behavior.