Which crystal structure do diamonds and silicon share?

The diamond cubic structure - each atom bonded to four neighbours at the corners of a tetrahedron, repeating in a rigid three-dimensional net. Silicon, germanium and grey tin have the same arrangement, which is why they behave as a family.

The structure explains diamond directly. Every carbon is locked by four strong covalent bonds with no free electrons and no planes that slide, which gives extreme hardness, no electrical conduction, and the highest thermal conductivity of any bulk material - a diamond feels cold because it pulls heat out of your skin faster than anything else will, which is the basis of the old test.

Graphite is the same element in sheets, and is thermodynamically the more stable form at room temperature, meaning every diamond is slowly turning into pencil lead. The conversion needs so much energy to get started that it will not happen in the age of the universe. Silicon's version of the same lattice is what makes semiconductors possible: the gap between its energy bands is small enough that deliberately added impurities can control conduction, and the entire electronics industry lives in that gap.