Carbon atoms can join in different ways. In diamond each carbon atom forms four strong covalent bonds in a giant structure, so diamond is very hard, has a very high melting point and does not conduct electricity.
In graphite each carbon atom forms three covalent bonds, making layers of hexagonal rings. There are no covalent bonds between the layers, so they slide and graphite is soft. One electron from each atom is delocalised, so graphite conducts electricity. Graphene is a single layer of graphite. Fullerenes are hollow shapes of carbon atoms.
Nanoparticles are 1 to 100 nm across and contain a few hundred atoms. They have a very high surface area to volume ratio, so they can behave differently from the same material in bulk.