the crystal system
There are not infinitely many shapes of unit-cell box — only seven basic families, sorted by how symmetry constrains their edge lengths and angles. These seven crystal systems are the top-level filing cabinet of all crystals, running from the most symmetric (cubic) to the least (triclinic).
The seven, by their axial relations: cubic (a=b=c, all angles 90); tetragonal (a=b, not equal to c, all 90); orthorhombic (a, b, c all different, all 90); hexagonal (a=b, not equal to c, alpha=beta=90, gamma=120); trigonal/rhombohedral (a=b=c, the three angles equal but not 90 — or described on hexagonal axes); monoclinic (a, b, c all different, alpha=gamma=90, beta not equal to 90); triclinic (all edges and all angles unequal, no constraints). For example, table salt is cubic, quartz is trigonal, gypsum is monoclinic.
The defining feature of each system is its characteristic symmetry, not merely the metric relations — a crystal is cubic because it has four 3-fold axes along the body diagonals, and that symmetry FORCES a=b=c, not the other way round. A cell that happens to have a=b=c by accident, without the symmetry, is not truly cubic.
Sort by symmetry, not by looks: quartz (a=b, not equal to c, gamma=120) is trigonal, not hexagonal, because its highest rotation axis is 3-fold, not 6-fold — even though its cell metric looks hexagonal. The system is named for the symmetry the atoms actually have.
Crystal system is decided by symmetry, and only then reflected in the cell shape.
There are seven crystal systems but fourteen Bravais lattices and thirty-two point groups — the counts differ because one system can host several centerings and several point groups. Do not equate system with lattice or with crystal class.