Azurite (also know as Chessylite) is copper carbonate hydroxide. Azurite grows as crystal nodules is often found in association with the green copper mineral malachite. The azurite come from the Arabic word for blue.
Malachite is a carbonate mineral. The stone's name derives (via Latin and French) from Greek 'molochitis' Malachite was used as a mineral pigment in green plants from antiquity until about 1800. It has also famously always been used as a decorative stone for ornamental purposes and inlay work.