Copper and graphite are two of the most commonly used electrode materials. Both have properties that make them excellent electrode material.
Copper has low wear burning property which makes for more consistent and precise geometries.
It is also compatible with most polishing circuits. The excellent structural integrity of copper is key to producing very fine surface finishes, even without use of special polishing circuits. Copper has good resistance to DC arcing in poor flushing situations.
Compared to graphite electrodes, copper is less efficient. It is difficult to machine or grind due to being soft and squashy. It is very difficult to de-burr. In fact, it is longer to de-burr than to manufacture.
Graphite is the most widely-used electrode material. It was first introduced five decades ago. It has very high melting point which makes it an ideal electrode material. In fact, graphite doesn't really melt. Graphite vaporizes from solid to gas. It removes material twice as fast as copper and also wears less.
There are different grades if graphite for different applications, from roughing to finishing. Some negatives to graphite is its low mechanical strength. Graphite is dirty. It produces plenty of dust which can really mess up the shop. Many European and Japanese shops prefer copper due to cleanliness. With graphite, poor flushing can produce surface anomalies in the EDM surface caused by DC arcing.