Those algae are symbionts — organisms that help their host (because it also helps themselves). These algae make food, which they'll share with the coral. When hot water makes the algae flee, the corals turn white and can eventually starve.
Symbiosis (from Greek σύν "together" and βίωσις "living") is close and often long-term interaction between two different biological species. In 1877 Albert Bernhard Frank used the word symbiosis (which previously had been used to depict people living together in community) to describe the mutualistic relationship in lichens. In 1879, the German mycologist Heinrich Anton de Bary defined it as "the living together of unlike organisms."