The brief summary is that Calcite is a crystal. This means that it is a solid that has an ordered structure of its atoms (like sugar crystals or diamonds) as opposed to "amorphous" "unstructured" or disorganized substances. Here's a good picture showing the order (quartz crystal) vs. disorder (amorphous glass) of molecules in each solid: calcite

So, Calcite is a crystal. It forms from the amorphous unstructured substance calcium carbonate. This amorphous calcium carbonate is called "ACC." ACC crystallizes into a substance called vaterite very quickly. When vaterite is present in water it dissolves and re-forms as Calcite. Calcite is more stable than vaterite so once it becomes Calcite it stops changing. The details of this are fairly complicated, but essentially this is the end of the line: ACC --> Vaterite --> Calcite.