Crystal form or structure is a description of the ordered arrangement of the atoms, ions or molecules in crystalline materials. Ordered structures occur due to the intrinsic nature of the constituent particles that form symmetrical patterns.
Crystal system is really classification of those patterns. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. A crystal family is determined by lattices and point groups.
Here is a short piece about those systems:
Lastly. There is some short reading about crystal habit: