Fresh copper shines reddish-orange. In air, however, the surface slowly reacts with oxygen and moisture – a process that can take years to decades.
Depending on the environment, different green copper compounds form: in cities and industrial areas, sulfur dioxide in the air often leads mainly to basic copper sulfate; near the coast, copper chloride tends to dominate. Textbooks often simplify the patina as copper carbonate, even though in practice that's often only part of the mixture.
What all these compounds have in common is that they form a layer significantly denser than the rust that forms on iron. Because the patina is so dense, it protects the copper underneath from further reaction – so the layer only keeps growing very slowly from then on. This is exactly why copper is deliberately used for roofs and statues: it builds itself a lasting, green protective shield.