Copper is one of the lower tier metal from Tinkers' Construct. You obtain it from oreberry, ore, gravel ore, or as loot from dungeons.
Copper is also a main ingredient in forging Bronze, a stronger metal than iron. To cast parts in a Smeltery, you must make Aluminium Brass (3 Raw Aluminium, 1 Copper). It is required to have at least tier 2 (stone) pickaxe/shovel to collect the various Copper Ores and Copper Gravel Ores scattered around the overworld.
Copper on it's own can be cast out into parts with the smeltery on a casting basin and a hardened gold cast for specific parts of tools and weapons.
Spawning
Copper is often found alongside iron or very near it in caves, or in a random pile of gravel on the surface world.