Highly ordered atomic arrangement - means that the atoms in a mineral are arranged in an ordered geometric pattern. This ordered arrangement of atoms is called a crystal structure. For each mineral has a crystal structure that will always be found for that mineral, i.e. every crystal of quartz will have the same ordered internal arrangement of atoms. If the crystal structure is different, then we give the mineral a different name. A solid compound that meets the other criteria, but has not definite crystal structure is a said to be amorphous.
One of the consequences of this ordered internal arrangement of atoms is that all crystals of the same mineral look similar. This was discovered by Nicolas Steno in 1669 and is expressed as Steno's Law of constancy of interfacial angles - angles between corresponding crystal faces of the same mineral have the same angle. This is true even if the crystals are distorted as illustrated by the cross-sections through 3 quartz crystals shown below.