Yes, Diamond is the hardest substance known. However, there often seems to be some confusion about what exactly that means.
Hardness refers specifically to a substance's resistance to abrasion. If you rub two substances together, the harder one will scratch the softer substance.
This does not refer at all to the density of the substance, or its tensile strength, or torsional strength, etc. etc. It only refers to its resistance to being scratched by other objects.
Diamonds can be cut because they have a crystalline structure. By applying careful pressure, the diamond can be fractured along these cleavage planes.
This crystalline structure also makes diamond unsuitable as a building material. A beam or column made of diamond would break along these crystalline cleavage planes, and could not support the weight of something like steel (which has a hardness of only about 7).
Density is also unrelated to hardness. Both gold and lead are very dense materials, but they are both very soft. They can be scratched or scored very easily.
The chemical formula for diamond is simply C. Pure diamond is nothing but carbon. In theory, once nanotechnology becomes common place, it should be very easy to produce as much diamond as we want as long as we have a source of carbon. In Neil Stephenson's novel Diamond Age diamond is more common that glass, forming most windows and such. For this to be safe, we'd probably need to create laminated sheets of diamond with each layer oriented a different way. That way a crack in the diamond's crystal structure would only be in that layer.
The refractive index of diamond is very high too, about 2.42 I think. This might make viewing through diamond quite distorted.