To start off with a mineral is a naturally occurring crystalline solid with a well-defined composition. Minerals themselves are not rocks, but together they form the constituents of rocks.
Azurite is personally one of my favorite minerals, and is a gorgeous deep blue copper carbonate (meaning it has the carbonate ion, CO32-). In terms of volume, carbonate rocks are the most significant precipitated rocks on Earth. Azurite is formed from the weathering of copper ore deposits (the copper will oxidize [losing electrons], combining with CO2 and H2O) and can be found world wide.
Azurite is frequently found with malachite, a vivid green copper carbonate, as azurite tends to be replaced by malachite in weathering.
On the Mohs scale of hardness (which ranges from 1 [talc] to 10 [diamond]), Azurite ranks from 3.5-4, meaning that it’s just about too hard to be scratched by a copper penny, but you would definitely be able to scratch it using a pocket knife. Since Azurite has a relatively low hardness, it’s mainly used as a gem and in jewlery.
(Images: left, specimen from the Lawrence University collection; right, specimen from the Field Museum in Chicago, IL – both images taken by myself)