Calcite is calcium carbonate, and by any measure of how much of the earth's surface it accounts for and how much of it was made by living things, it is the most consequential mineral there is.
Limestone is calcite. So is chalk, which is calcite in the form of the microscopic plates of coccolithophores that settled on a Cretaceous sea floor in numbers beyond counting. Marble is limestone recrystallised by heat and pressure. Stalactites and stalagmites are calcite deposited from solution. Most shells, most coral skeletons and a great deal of the sediment on the ocean floor are calcite. The mineral is the principal reservoir in the carbon cycle over geological time.
It is identified in seconds. Calcite fizzes in dilute acid, releasing carbon dioxide, and it has a hardness of three, meaning a copper coin will scratch it and a steel knife certainly will. Those two tests separate it from almost everything it might be confused with.
That softness and solubility govern everything built from it. Limestone and marble buildings dissolve slowly in rain, faster in acidified rain, and the weathering of carved detail on a medieval cathedral is calcite going into solution. Karst landscapes, sinkholes and cave systems are the same process at scale.
Its optical behaviour, the strong double refraction that makes a line viewed through a clear rhombohedron appear twice, is treated in the other article on this stone. It is worth noting only that the optical property and the geological importance belong to the same crystal structure.