Graphene consists of carbon atoms arrayed in a hexagonal pattern in a large sheet. Layers of graphene constitute graphite, the familiar "lead" in pencils. Nanotubes are, in a sense, tubular graphene.
Electrons can very easily travel along the carbon sheet making graphene an extremely good electrical conductor. This conductivity suggests that graphene could have a future in the semiconductor industry, with the fabrication of high speed transistors. See an image of graphene with pi orbitals explaining this.