the graphite structure
The graphite structure is the arrangement of carbon in pencil lead: flat sheets of carbon atoms tiled into hexagons, like endless chicken wire, loosely stacked one above another. It is the layered opposite of diamond, and comparing the two is the cleanest illustration of how the same atoms in different structures can be wildly different materials.
Within a sheet, each carbon is covalently bonded to three neighbours (sp2 bonding) at a short distance of 1.42 angstrom, giving strong, planar, three-coordinate networks. Between the sheets, however, there are only weak van der Waals forces, and the layers sit a full 3.35 angstrom apart, more than twice the in-plane bond length. The common form (hexagonal graphite) stacks its layers in an ABAB sequence. So graphite is strongly bonded in two dimensions and barely bonded in the third.
That anisotropy explains everything graphite does. The weakly held layers slide over one another easily, which is why graphite marks paper, lubricates, and feels slippery; the delocalised in-plane electrons let it conduct electricity along the sheets but poorly across them. Graphite and diamond are allotropes, the same carbon atoms in profoundly different structures, and a single graphite sheet, peeled off, is graphene, the founding two-dimensional material.
A pencil writes because van der Waals-bonded graphite layers, 3.35 angstrom apart, shear off and stick to the paper while the strong in-plane bonds keep each flake intact.
Strong in-plane, weak between planes: two-dimensional bonding written into a three-dimensional crystal.
Graphite is not weakly bonded overall; the in-plane covalent bonds are among the strongest known. It is the huge contrast between strong in-plane and weak between-plane bonding that gives graphite its softness and slipperiness.