A heatsink has one side that is flat and contacts the processor package, and the other side is usually a group of long, thin fins or pegs. This is to get the maximum surface area, and best thermal conductivity with the air around the chip. The vast majority of heatsinks are aluminum. The reason for this is that aluminum has a decent thermal conductivity. It's also light, not very expensive, and is easy to work with.
Copper is much heavier, and can hold a lot more heat.
It is also harder to work with. Silver heatsinks are rare due to the cost of silver, but they do exist. Some designs use a copper plating to suck heat away from the processor quickly, and then use aluminum for the rest of the heatsink. Well, you typically have some sort of metal clamp packaged with your heatsink, but then if your processor is just a tiny bit off from perfect flatness, and your heatsink as well, your processor may not contact the heatsink directly. That means that air is cooling your processor, and air does not cool well on a small flat surface.