The main formal oxidation state of Carbon is 4+ (there are other oxidation states, but they use carbon’s valence electrons in bonding) and the electronic configuration is [He]2s22p2.
Forming millions of compounds both inorganic and organometallic, Carbon plays a unique central role in organic chemistry and can be considered a building block of our world.
Carbon has two main isotopes with relative abundances: 12C (98.9%, I=0), and 13C (1.1%, I=0.5). I is the nuclear spin and the half-integer value of the nuclear spin for 13C gives it its usefulness in structure determination by NMR.
Elemental carbon occurs in several different forms or allotropes. The most abundant forms are diamond and graphite with fullerenes rare on Earth. They exhibit markedly different properties due to their very different structures.
Diamond has a 3D lattice crystal structure and is the hardest known substance because of strong C-C covalent bonds. Each Carbon atom forms four bonds tetrahedrally arranged to other Carbon atoms. This results in an open but very strong 3D structure.
Graphite is an electrical conductor with a layered lattice crystal structure. It is slippery and used as a lubricant (due to a lubricating effect coming from the ability of layers to slide over one another, as they are weakly held together by van der Waals forces). In graphite, each Carbon atom forms three covalent σ-bonds to further Carbon atoms. σ-bonds are made of sp2 hybrid orbitals. The remaining p-orbitals, which are perpendicular to the plane of the σ-bonds overlap to form a delocalized π-system. The weak van der Waals forces hold together the widely separated planes.
Fullerenes including Carbon 60, 70, 76 and 84 are formed by replacing some of the six-membered rings by five-membered rings of graphite (like bent graphite sheets). In laboratories, Carbon fullerenes are formed when an electric arc is struck across graphite electrodes in an inert atmosphere. Carbon 60, often referred to as Buckminster-Fullerene is the most abundant, but others such as C70, C76 and C84 occur and are produced naturally in smaller quantities.
An individual Carbon 60 molecule is shaped like a soccer ball and has icosahedral symmetry. Each pentagon is surrounded by hexagons and each hexagon is surrounded by three pentagons and three hexagons. It crystallizes to give a magenta solid and dissolves in benzene giving a magenta solution. The 13C NMR spectrum shows one signal, meaning all of the Carbon atoms in Carbon 60 are equivalent.
An individual Carbon 70 molecule has the structure of Carbon 60 with an extra strip of five hexagons around the center of the soccer-ball. When crystallized, it is a red-brown solid and dissolves in benzene to give a red solution. The 13C NMR spectrum shows five signals, so Carbon 70 contains Carbon atoms in five different environments.
The reactivity of fullerenes is somewhere between that of an arene (with an extended graphite like p-system) and an alkene (with an isolated C=C double bond).
The addition product with K is a superconductor below 18K, whose structure is a face-centered cubic array of C60 molecules, with K+ ions occupying all the octahedral and tetrahedral holes; the product with OsO4 is a standard alkene-like addition, where the OsO4 adds across a C=C bond. The reaction with Na and NH3 is known as the Birch reduction.
Here is some great further reading:
Reactivity of Fullerenes and the Birch reduction