This is a way of
classifying the
symmetry of objects and
lattices of objects. The symmetry of any such object will correspond to a particular point group. The group defines the
symmetry operations that the arrangement exhibits.
For lattices] such as in crystals (with long-range order) there are only 32 different points groups. An additional 12 non-crystalline point groups define symmetries that are only exhibited if there is no long-range order (these are things like infinite and 5-fold rotational axes).
Though originally developed as an intellectual exercise (with no apparent practical application) Point groups are essential to the understanding and interpretation of things like X-ray Crystallography and Vibrational Spectroscopy.