There are corals that are grown in captivity. They are generally labeled aquaculture.
Corals in the wild can be quite colorful especially when placed under strong blue lighting which causes certain corals to fluoresce.
Corals reproduce in two ways, I believe. One is just by simple growth. A part or frag of the coral can be removed, placed in a decent environment to make a new colony. These new colonies are identical to the old ones. I think you can get a rare sport by this method. But you will get nothing like the variety you can get by sexual reproduction.
Corals can reproduce sexually. Corals spawn at certain times of the year and under certain conditions. The corals release clouds of male and female gametes that find each other and fuse. The new coral larvae are planktonic for some time before they settle on the reef to form new colonies. I don't know that anyone has ever successfully spawned corals and gotten new colonies by doing this.
Most coral producers, frag their corals. They buy wild caught colonies, grow them and frag them. Obviously, wild corals that are vividly colored command a premium price. Of course, irresponsible collecting can deplete native stocks. Some areas handle their coral resources more responsibly than others. I think in Fiji, they often collect just frags of the wild caught corals. They place these selected frags in ocean farms and grow them to colonies, mariculture. They sell the resulting colonies to the hobbiest market. Obviously, they can collect colorful frags and sell colonies on a value added basis.