Carbon crystallizes in two forms: diamond and graphite. How can these very unlike things both be crystalline carbon? It's the way the atoms are arranged. In a diamond, each carbon atom is linked to four more creating a very strong crystal. The carbon in graphite link together and form sheets which is why graphite is so slippery.
Diamond forms in the cubic crystal system which manifests in a number of growth forms. The following rendering shows the typical arrangement of carbon in a diamond crystal. Each cube represents the basic carbon structure pictured above.
Octahedron: The most common form the diamond crystal assumes. This mimics the rendering above. Generally suitable for finishing.
Dodecahedron: Similar to the octahedron, but without the points. Look at the rendering and remove the corners. Also generally suitable for finishing.
Cube: Again, look at the rendering and you will see that if you clip off the points more severely than the dodecahedron, you are left with a cube. The vast majority of these are put to industrial use.
Sometimes a crystal gets turned around, so to speak, while forming. Have a look at that diamond octohedron above. Slice it in half along a diagonal and twist it 180 degrees. You now have a macle. Many of these wind up finished in triangular shapes.
Growth conditions can cause uneven crystal face development. Large diamond crystals are often strange shapes, like this 507 carat rough crystal:
It sold at auction in 2010 for $35 million.