Adoption was a procedure, often legal but sometimes informal, by which a person or persons took an orphaned or abandoned child in and raised them as their own. Informally term applied to the acquisition of both objects (i.e. a pet) or places (i.e. a colony) as adoption.
In 2151, Sub-Commander T'Pol advised Captain Jonathan Archer that it would be wise to adopt Vulcan standard protocol not to interfere with a pre-warp society when a first contact with the Akaali was discussed. (ENT: "Civilization")
Keyla adopted her dog Rhylo when she was a puppy. (ENT: "Two Days and Two Nights")
During their childhood, Rom's brother Quark told everyone he was adopted in order to explain the younger boy's small lobes. He recounted this to Martus Mazur in 2370, but the El-Aurian wasn't interested. (DS9: "Rivals")
After his mother's death, Rugal, a Cardassian war orphan, was adopted by Proka Migdal. (DS9: "Cardassians")
The first draft script for Star Trek III: The Search for Spock included a male Vulcan named Galt who had been adopted by Sarek.