The carbonate family of gemstones is a large family of minerals which many occur as secondary minerals formed during weathering or the un-rocking of the rock. Most dolomite is a secondary formation made by reaction of calcite with magnesium-rich water during weathering. Magnesite, (MgCO3), a related carbonate, forms as an alteration product of mafic igneous rocks rich in iron and magnesium.
Carbonates are commonly formed in oxidizing weathering environments.
Most carbonates are softer and acid sensitive. This is not a complete list. There are many many carbonate minerals.
Aragonite - Calcium Carbonate is a low temperature, near surface mineral that in addition to calcite, forms speleothems in caves, massive lamellar deposits by geysers and hot springs, as seafloor oolites, is a component of sea shells and as a replacement mineral. It was named in 1797 for the type locality, Molina de Aragon, Spain.
Dolomite - Calcium Magnesium Carbonate - (see page about Dolomite for more info)
Magnesite - Magnesium Carbonate - Occurs as an alteration product of magnesium rich stones, often when magnesium-rich rocks come into contact with carbon dioxide-rich water. Magnesite usually appears as a dull chalky white, but can also be found in gray, brown, yellow, orange, pale pink and colorless varieties too. These properties make it perfect to dye and pass off as other stones like turquoise.