1. Diamond: Diamond is the hardest naturally occurring mineral, known for its brilliance and high refractive index. Its structure is a rigid, three-dimensional network of carbon atoms arranged in a tetrahedral lattice.

2. Graphite: Graphite is a soft, black, and flaky material with excellent electrical conductivity. Its structure is layered, with each layer consisting of a hexagonal network of carbon atoms. The layers are held together by weak van der Waals forces, allowing them to slide easily past each other.

3. Fullerene: Fullerenes are a class of carbon molecules with a closed, cage-like structure. The most famous example is buckminsterfullerene (C60), which has a spherical shape resembling a soccer ball. Other fullerenes have different shapes and sizes, but they all share the characteristic of being hollow and cage-like.

There are other forms of carbon like amorphous carbon, carbon nanotubes, and graphene, but these three are often considered the most common and well-known.