Copper is a found in or around almost every cave in the game, and can be mined with any pickaxe. Just like any other ore, to refine it, you must light a flammable object, and place the ore next to the flaming object. After a few seconds, it will turn into refined copper, and can be crafted into tools or bronze.
It is used in numerous tools, such as the Copper Knife, Copper Pickaxe and Copper Sword. It's main use however, is creating bronze, a much more effective material for creating tools. As a result, most typical players skip copper and go straight to bronze.