These planes of relative weakness are a result of the regular locations of atoms and ions in the crystal, which create smooth repeating surfaces that are visible both in the microscope and to the naked eye.
[1] The classic example of cleavage is mica, which cleaves in a single direction along the basal pinacoid, making the layers seem like pages in a book.
In graphite, carbon atoms are contained in layers in a hexagonal pattern where the covalent bonds are shorter (and thus even stronger) than those of diamond.
So weak is this bond that it is broken with little force, giving graphite a slippery feel as layers shear apart.
Synthetic single crystals of semiconductor materials are generally sold as thin wafers which are much easier to cleave.
Simply pressing a silicon wafer against a soft surface and scratching its edge with a diamond scribe is usually enough to cause cleavage; however, when dicing a wafer to form chips, a procedure of scoring and breaking is often followed for greater control.