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This is black-band disease, which is the only treatable disease seen in coral. It is caused by the bacterium Phormidium corallyticum and is a progressive disease, starting at a point and spreading outward. It is called "black-band disease" because of the clear line of black that marks the disease's progression, surrounding the exposed skeleton of dead coral. Other diseases include white-band disease, yellow blotch disease, red-band disease, and coral bleaching, which is probably the best known of them all and involves the loss of the algae that normally live in coral tissue.
Coral is very sensitive to environmental changes, including salinity, water clarity (turbidity), pH, temperature, and also the amount of water above them. Coral diseases, including black-band disease, have been becoming more common in recent years due mainly to global warming and tourism. Global warming has lead to changes in water temperature and other conditions, while tourism increases water turbidity, and can introduce foreign toxins and bacteria, among other things. Unfortunately, the world's coral reefs are not doing very well. Though able to easily recover from sort-term disasters such as hurricanes, long-term exposure to negative conditions can kill even the largest of reefs.
Coral reefs are a very important ecosystem, providing homes and food for a very large number of other species, including acting as nursery habitat for many. They also protect coastlines from such things as erosion and act as indicators of the state of ecosystem. Throughout history, they have also acted as important sources of food to humans through the animals that live on and around the coral and also support many communities through tourism. Hopefully, we can maintain the world's reefs and keep any more from dieing out.
Sources are NOAA, the EPA, and the Florida Museum of Natural History.