I guess I'm a little confused too. There are 3 common polymorphs of CaCO3 produced in calcifying organisms: high-Mg calcite, low-Mg calcite and aragonite. Corals produce almost entirely aragonite, perhaps with some calcite as well (though these are final products--amorphous CaCO3 and maybe vaterite are probably the initial phases which eventually reorganize to aragonite).
If you're intersted in the process of biomineralization, we can discuss it here, but note that there are literally entire symposia and many books dedicated to the subject. Even so, significant unknowns still remain in spite of work done to date.
Chris