Diamond, graphite, fullerenes and graphene are all forms of carbon, but their structures and bonding give them different properties. In diamond each carbon atom forms four strong covalent bonds to four others in a rigid giant structure. Breaking this takes a lot of energy, so diamond is very hard and has a very high melting point. All four outer electrons are used in bonding, so there are no free electrons and diamond does not conduct electricity.
In graphite each carbon atom forms three covalent bonds with others in the same flat layer, in a pattern of hexagons. This leaves one outer electron on each atom that is delocalised, which means it is free to move, so graphite conducts electricity. The layers are held together only by weak intermolecular forces, so they slide over each other and graphite is soft and slippery. It still has a high melting point because the covalent bonds within the layers are strong.
Graphene is a single layer of graphite, one atom thick. It is very strong, light and conducts electricity. Fullerenes are molecules of carbon shaped as hollow cages or tubes. Carbon nanotubes are very strong and conduct electricity, and fullerenes can be used to carry drugs or as catalysts.
