Copper turns green because of chemical reactions with the elements.
For the same reason that iron rusts.
Just as iron that is left unprotected in open air will corrode and form a flaky orange-red outer layer, copper that is exposed to the elements undergoes a series of chemical reactions that give the shiny metal a pale green outer layer called a patina.
The patina actually protects the copper below the surface from further corrosion, making it a good water-proofing material for roofs (which is why the roofs of so many old buildings are bright green).
In fact, the weathering and oxidation of the Statue of Liberty's copper skin has amounted to just .005 of an inch over the last century, according to the Copper Development Association.