An adult, also known as a grown-up or, on Miri's planet, a Grup, was a fully-matured lifeform.
In 2083, nearly half of the adults of the colonists of the SS Conestoga died, the other half was sick and only the children seemed unharmed. The Novans called their adults as "grown ones", as Nadet recalled when she told that Humans gutted their grown ones. (ENT: "Terra Nova")
Upon reaching adulthood, Ferengi were allowed to earn profit and have their own home. They called the age at which they reached adulthood their Age of Ascension. (DS9: "Family Business")
When Ferengi invaded the USS Enterprise-D, they began abducting all the able-bodied adults on the ship, however, Jean-Luc Picard, Ro Laren, Keiko O'Brien, and Guinan were present, having been temporarily transformed into children due to an accident and they, with the help of Alexander Rozhenko and William T. Riker managed to outwit the Ferengi and transport them away, saving the rest of the crew. Later, Picard, Ro, Keiko, and Guinan became adults again when the other crew members ran their adult DNA patterns through the transporter. (TNG: "Rascals")
Lwaxana Troi sometimes referred to her daughter Deanna Troi as "little one", even into Deanna's adulthood, which annoyed Deanna as she thought it was disrespectful. (TNG: "Ménage à Troi")
The lifeform that manifested itself as Clara Sutter's imaginary friend Isabella believed adult humanoids to be serious and immoral due to the fact that they disallowed their children to do certain things, however, Picard explained that humanoid adults only did that because they wanted to protect the children. (TNG: "Imaginary Friend")
Clara Sutter believed adults to be more serious than children, due to the fact that most did not take Isabella seriously, but when Guinan did, she believed not all adults were more serious than children. (TNG: "Imaginary Friend")
In most humanoids, adulthood was reached at around eighteen years of age, however, Klingons and Jem'Hadar reached adulthood earlier. (DS9: "The Abandoned") A Changeling reached adulthood when it was able to shapeshift consistently for significant periods of time and had reached its maximum size. (DS9: "The Begotten")
Minors were not allowed in Ten Forward or engineering unless accompanied by an adult. (TNG: "Imaginary Friend")
On one planet, a plague had infected all the adults, but the children were immune. The plague changed the adults' facial appearances, made them destructive, and ultimately killed them, and the children never matured and became wary of all adults. (TOS: "Miri")
Vulcans experienced pon farr every seven years of their adult lives. (ENT: "Fusion", "Bounty"; TOS: "Amok Time"; Star Trek III: The Search for Spock; VOY: "Blood Fever", "Body and Soul")
Seven of Nine was assimilated by the Borg as a child. As such, she had no adult memories prior to being Borg and felt less compelled to get her individuality back than Marika Wilkarah, P'Chan, and Lansor (who did have pre-assimilation adult memories) when the four of them were stranded on Planet 1865-Alpha. (VOY: "Survival Instinct")
Adult Borg usually could not survive without a cortical node. Borg children and adolescents could potentially survive without one, but it was still risky. (VOY: "Imperfection")
Miles O'Brien spent the entirety of his adulthood serving in Starfleet and was considered a loyal officer the whole time. (DS9: "Tribunal")