Graphene is an allotrope of carbon. Its structure resembles a single layer of graphite. Graphene has a very high melting point. It is very strong because of its large regular arrangement of carbon atoms joined by covalent bonds. Like graphite, graphene conducts electricity well because it has delocalised electrons that are free to move across its surface.
A nanotube resembles a layer of graphene, rolled into a tube shape. Nanotubes have high tensile strength, so they are strong in tension and resist being stretched. Like graphene, nanotubes are strong and conduct electricity because they have delocalised electrons.
Nanotubes can be added to other materials, for example in sports equipment, to make them stronger.
Buckyballs are spheres or squashed spheres of carbon atoms. They are made up of large molecules so are not classed as giant covalent structures. Weak intermolecular forces exist between buckyballs. These need little energy to overcome, so substances consisting of buckyballs are slippery and have lower melting points than graphite or diamond.
The ball structure means that this type of fullerene structure can be used to carry small molecules. This can be used to carry medical drugs into the body.