1530s (but not common until mid-17c.) "grown, mature," from Latin adultus "grown up, mature, adult, ripe," past participle of adolescere "grow up, come to maturity, ripen," from ad "to" (see ad-) + alescere "be nourished," hence, "increase, grow up," inchoative of alere "to nourish," from a suffixed form of PIE root *al- (2) "to grow, nourish."
Meaning "mature in attitude or outlook" is from 1929. As a euphemism for "pornographic," it dates to 1958 and does no honor to the word. In the old British film-rating system, A indicated "suitable for exhibit to adult audiences," and thus, implicitly, unsuitable for children (1914).