According to Jesus de La Fuente, CEO of Graphenea:
Graphene is, basically, a single atomic layer of graphite; an abundant mineral which is an allotrope of carbon that is made up of very tightly bonded carbon atoms organised into a hexagonal lattice.
He also states that there are three commonly-known naturally occuring allotropes of carbon - Diamond, Graphite, and amorphous carbon. Graphene's structure is similar to that of Graphite, but Graphene itself is much different because it has a $sp^2$ hybridization, compared to $sp^3$ hybridization of carbon in the form of Diamond, and Graphene has a thickness of just 0.345Nm (nanometers). This means that it does not form tetrahedrals like common carbon atoms, making Graphene similar but special compared to other forms of carbon.
Also stating Jesus de La Fuente in Graphite-Graphene Comparison:
Graphene is fundamentally one single layer of graphite; a layer of $sp^2$ bonded carbon atoms arranged in a honeycomb (hexagonal) lattice. However, graphene offers some impressive properties that exceed those of graphite as it is isolated from its ‘mother material’.
To elaborate, Graphene has special properties that set it apart. There are similarities: stacking layers of Graphene may be similar to creating Graphite; creating a rope of Graphene may be similar to creating Crabon Nano-tubes; both have $sp^2$ hybridizations. However, Graphene has special properties that make it stronger when stacked because of how it is extracted and isolated.
As for the conductivity, Jesus de La Fuente states that it's electronic properties is what sets it apart (See Graphite-Graphene Comparison). Fundamentally, Graphene is a zero-overlap semi-metal, with both holes and electrons as charge carriers, that has very high electrical conductivity by itself. Graphite carbon allotropes have 4 electrons in it's valence shell available for bonding, however, Graphene has 3 electrons that are bonded with other Graphene atoms in the 2D Plane (a.k.a. other Graphene atoms next to it on the same one-atom thick sheet). This makes a single sheet very conductive, because the remaining electron that isn't bonded with other Graphenes in the same sheet are in the 3D Plane (above and below the sheet), causing what is called, pi electrons. However, when Graphene sheets are stacked together, their pi electrons bond with each other, thus reducing the electric conductivity of the Graphene.