CARBON. The element that makes diamond strong is the carbon found inside it. You can actually make substances stronger than diamond by using carbon in different patterns and arrangements of its molecules.
On the "MOHs" scale, diamond is the hardest stone at 10, and talc the softest at 1. As others have noted here, the only thing which can cut a diamond is another diamond, hence the folk saying, "diamond cut diamond". For an interesting discussion of the MOHs table, go to http://en.wikipedia.org/wiki/Mohs_scale_of_mineral_hardness