Hard Corals are the building blocks of Coral Reefs.
Coral Polyps form Coral Colonies that group together under certain conditions to form Coral Reefs.
It is the hard coral skeletons created from calcium carbonate (limestone) secreted by Coral Polyps that form the 'Hard Coral'. The skeletons When most people refer to corals, it is the hard coral they are referring to.
Soft Corals do not have a hard skeleton, and are soft and bendable with a soft or leathery feel.
These corals often have wood-like cores made of calcium (spicules) for support and fleshy rinds for protection. Known as non-reef building corals (ahermatypes) they do not always have the relationship with algae (zooxanthellae) like hard corals.