Allotropes are different structural forms of the same element in the same physical state.

For example, diamond, graphite and graphene are allotropes of each other because they are all solid carbon.

1. Carbon atoms are arranged in layers.
2. The forces between layers are weak intermolecular forces.
3. This means they don't need much energy to break, so layers slide over each other easily.

1. Soft and slippery.
2. Conducts electricity along layers.
3. Lower density than diamond.
4. High sublimation temperature but lower melting point than diamond.

Graphite can conduct electricity because it has delocalised electrons that can flow and carry charge.

Graphite has a giant covalent lattice structure.

Each carbon is covalently bonded to 3 other carbon atoms, forming a sheet of hexagonal rings. The separate sheets are held together by intermolecular forces. It also has delocalised electrons that can flow throughout the structure. 

3

Diamond can't conduct electricity because all of carbon's outer electrons are involved in bonding so there are no free electrons to carry charge.

Diamond has a giant covalent lattice structure.

Each carbon is covalently bonded to 4 other carbon atoms in a tetrahedral arrangement.

4

The carbon atoms at the edge haven't had all their bonds drawn - look at the atoms in the middle to see the 4 bonds.

Giant covalent structures are an extensive interconnecting networks of covalently bonded atoms, involving huge lattices extending in three dimensions.

Buckminsterfullerene

Buckminsterfullerene has the molecular formula C₆₀.

Fullerenes are spherical or tubular molecules of carbon atoms.

They have many strong covalent bonds between atoms which require a lot of energy to break.