Is carbon an insulator or a conductor?

It depends on the type, or allotrope, of carbon you're talking about. For a material to conduct electricity, it should have free electrons to carry the electrical current. In diamond, an allotrope of carbon, there are no such free electrons. This means that diamond does not conduct electricity. However, graphite, another allotrope of carbon, conducts electricity very well because it has an entire "sea" of free electrons.