Graphite conducts electricity because it possesses delocalized electrons in its structure. The honeycomb layout of the stacked carbon atoms of graphite leaves a single electron unbound in each hexagon. Each of these electrons is free to move within the structure, enabling electrical conduction.
The delocalized electron of graphite is indicated as a ring in the center of the C6 structure. This delocalization results in an equidistant bond length between all six bonds in each honeycomb cell of the graphite structure. If the bonds were not equidistant, the electron would be bound to the two atoms with the shortest bond length. Graphite is not the only allotrope of carbon that possesses a delocalized electron. C60 and graphene are also forms of carbon that possess this electron delocalization. Graphene is especially interesting because this delocalized electron is confined to a single plane or a few planes of atoms. This lets it surf across the electrical potentials of the carbon nuclei with little to no resistance, resulting in near-infinite conductivity. Although diamond is also an allotrope of carbon, it does not conduct electricity because the four electrons of each carbon atom are localized in pure covalent bonds with the four nearest neighboring atoms.