The main thing is that they are teeming with life. They are one of the richest habitats found on Earth. Loads of species of fish, sponges, jellyfish, crabs, shrimps, lobsters, sea snakes, octopus, clams, clown fish, eels, snappers, sharks, nudibranchs, sea cucumbers, starfish and marine plants live there. And that's just the tip of the iceburg. It would take pages to list all the life forms that are supported by coral reefs.
So much life can survive in corals because they provide sheltered warm, clean and clear watery environments, rich in food chains. The reef itself is built from coral polyps ~ which themselves are animals. These small creatures live in large colonies and when they die, their leave the hard branched structure that holds the reef together.
Growth of the reef is slow ~ about an inch per year, so any damage is catastrophic. A major global project is trying to coordinate efforts to protect coral reefs and stop the species that live their from becoming extinct before we even discover them. A recent project on a coral reef in Papua New Guinea found over 50 completely new species. What a waste if it was to be lost.