Diamond, graphite, buckminsterfullerene and relatives, carbon nanotubes (which come in countless varieties), activated carbon, and carbon ropes all have the same chemical formula--C. Carbon is unique in having such a vast array of solid forms. The reason it has this many forms is the same reason that it is the basis element of organic life: it is the smallest element with four valence electrons.
Since carbon is small, it forms strong bonds with other elements, including itself. The simple reason for this rule of chemistry is that the carbon nucleus holds its outer shell of electrons tightly since they are close to the nucleus. When it bonds with another atom, that atom's electrons are also held tightly to the carbon nucleus since they are close.
Since carbon has four valence electrons, it can bond with up to four other atoms (sharing one electron a piece). If it bonds with four other carbon atoms, then you have the basis for the diamond structure. Imagine four carbon atoms located symmetrically in 3-dimensional space around a central carbon atom. Those four atoms form a shape known as a tetrahedron. If you treat each of those four atoms as the center of another atomic tetrahedron, you will end up making the diamond crystal--eventually all the carbon atoms (except the ones on the edges of the crystal) will have four symmetric bonds around them. Interestingly, the silicon used to make computer chips is identical to diamond except silicon atoms replace the carbon atoms. Like carbon, silicon has four valence electrons. However, silicon isn't nearly as strong as diamond because...its outer electrons are an extra shell removed from the nucleus. By the way, when carbon (or silicon) bonds in this way, it is known as sp3 hybridization (because the spherical s orbital and the 3 dumbbell p orbitals merge into 4 symmetric tetrahedral orbitals).
In graphite, carbon only bonds with three other carbon atoms in a plane. This is sp2 hybridization--the s orbital merges with two p orbitals and the remaining p orbital is left intact. The three bonded carbon atoms form a symmetric triangle, as you probably would expect, around the center carbon atom. If you bond carbon atoms to those surrounding atoms in the same way, you'll form a hexagonal lattice known as a graphene sheet. Keep in mind this sheet is totally 2-dimensional. However, every carbon atom has the remaining p orbital sticking up. Those p orbitals are what allows two graphene sheets to bond together. A stack of graphene sheets is called graphite. The p-orbital bonds aren't that strong, so the sheets easily slide across eachother. That's why graphite can be used as a lubricant or a pencil "lead." Roll up graphene sheets and you get carbon nanotubes. These can have either a single sheet or many sheets and were only discovered in 1991 by a researcher Iijima at NEC.
Activated carbon is extremely important in filters (I have one on my sink at home). To make activated carbon, you take a source of carbon with lots of other junk in it, like peanut shells or charcoal, and you steam-heat it at very high temperature. All the other junk leaves, and you're left with a very porous, amorphous carbon solid. All those pores help trap gook (especially if the gook is carbon-based, like bacteria), giving you clean water with small dissolved minerals.
The most interesting forms of carbon are the fullerenes and related structures. These are gorgeous 3-d enclosed geometrical structures. Who would have thought that nature could create the soccer ball structure of buckminsterfullerene based solely on the simple electric force? And, by the way, that's all chemistry is--one atom interacting with another because of Coulomb's Law. Beautiful!