The carbon–carbon covalent (electron-sharing) bond is stable and strong.

The covalent bond is directional; there's a particular favorable structural orientation, and the atoms tend to arrange in this orientation and stay there.

In contrast to graphite, diamond comprises a 3D network structure (called, hey, diamond cubic), so the above characteristics apply in all directions.

Dislocations (i.e., 1D defects that direct stresses to break only a few bonds at a time, enabling slip) don't readily move in diamond's 3D structure as they do in ductile metals with more accessible [slip systems](https://en.wikipedia.org/wiki/Slip_\(materials_science\)). Thus, increasing the strain energy stored in a diamond leads not to [plasticity](https://en.wikipedia.org/wiki/Plasticity_\(physics\)) (softness) but to brittle fracture—Nature essentially directs this energy into creating new surfaces.

Carbon forms a very strong bond to itself when in diamond form as the structure is a tetrahedronal lattice. This is unlike carbon in graphite form where the bonding structure is sheet like.
The heat and pressure of diamond formation is what causes the additional bonds to form

Hardness is a measure of a material's property to resist local deformation, and is commonly implied to mean scratch hardness, or the ability to withstand abrasion / gouging along a surface.

Diamond's carbon structure - wherein it contains more atoms per unit of volume than any other known material, packed tightly into a tetragonal unit cell resistant to plastic deformation in every direction - renders it uniquely able to repel abrasion from anything else. Simply put - it's as hard as it is because of the way its carbon atoms are bonded.

Hardness is not the same as durability / brittleness or compressive strength, as a diamond can be shattered when struck or compressed with sufficient force.

Other people have answered why diamond is so hard.

But you asked also "what is hardness?"

This is usually defined by Mohs Scale. Which ranks materials based on their ability to leave a scratch in another material. Diamond will leave a scratch in most things. But if you try to scratch diamond the material you use in the attempt will come off worse.