Elemental carbon has basically two crystalline forms: planar and cubic. Cubic is that found in diamonds. Planar is the form found in graphite, and the "carbon" used in composites is really graphite. There are also other exotic, man-made forms of carbon like buckeyballs or nanotubes.
Graphite fibers make good structures due to their low density, high strength and stiffness. Their primary drawback is that they normally only exist as tiny fibers. A skinny fiber is great in tension, but lousy in compression. To keep the fibers aligned in the direction that they are best at taking loads, they must be embedded in a resin matrix- usually epoxy.
So ShiGuy, to put it in "physic-y" terms you can understand: "You can't push on a string"
"Q: How do you make a small fortune in racing?
A: Start with a large one!"