Domestication was the adoption and breeding of a species of wild animal to enable it to form a relationship with people, either for livestock or companionship purposes. The term could also refer colloquially to a person settling into married life. The term housebreaking referred to making an individual suitable for living inside the home, as opposed to outdoors, while taming or breaking referred to eradicating feral tendencies of an animal, such as a horse.
In 2154, after Jonathan Archer expressed surprise that T'Pol had ever had a sehlat as a pet, T'Pol confirmed that hers was domesticated and smaller than the one they'd just encountered in Vulcan's Forge. (ENT: "The Forge")
In 2258 of the alternate reality, Leonard McCoy chastised Spock for marooning James T. Kirk on Delta Vega, saying, ""Back home we have a saying: if you're gonna ride in the Kentucky Derby, you don't leave your prize stallion in the stable."" Spock countered the doctor, noting, ""A curious metaphor, doctor, as a stallion must first be broken before it can reach its potential."" (Star Trek)
In 2285, just prior to applying Ceti eels to Clark Terrell and Pavel Chekov, Khan Noonien Singh described them as pets that were not quite domesticated. (Star Trek II: The Wrath of Khan)
In 2372, Chakotay doubted that a monkey found on New Earth could be easily domesticated. Believing they would be remaining on the planet until their deaths, Kathryn Janeway replied they had the rest of their lives to try. (VOY: "Resolutions")
In 2375, shortly after the wedding of the Silver Blood USS Voyager's "B'Elanna Torres" and "Tom Paris", "Harry Kim" asked "Paris" if he was hungry. "Paris" declined joining him for a meal, saying he had dinner with "B'Elanna". "Kim" then quipped, "Married one day and you're already domesticated." (VOY: "Course: Oblivion")
In 2378, Q noted that Kathryn Janeway had successfully house-broken (civilized) Seven of Nine. (VOY: "Q2")