the seven crystal systems
Every possible box you could use as a unit cell can be sorted by the shape of the box — much as you might sort cardboard boxes by whether their edges are all equal and their corners are square. When you do this carefully, there turn out to be exactly seven distinct box shapes, called the seven crystal systems.
The sorting rule is the relationship among the three edge lengths a, b, c and the three angles alpha, beta, gamma. Cubic has a = b = c with all angles 90 degrees (the most symmetric). Tetragonal has a = b not equal to c, all 90 degrees. Orthorhombic has three unequal edges, all 90 degrees. Then come hexagonal, rhombohedral (trigonal), monoclinic, and finally triclinic, which has no equal edges and no right angles (the least symmetric).
This matters because the system fixes which directions in the crystal are equivalent, and therefore how strongly properties depend on direction (anisotropy). Engineering metals live mostly in the cubic and hexagonal systems; the low-symmetry systems are filled out by more complex minerals and compounds.
Cubic (a = b = c, all angles 90 degrees) is the most symmetric — salt, copper, iron. Triclinic (all edges and angles different) is the least — many feldspar minerals. The other five sit in between.
Seven box shapes, from the perfectly square cube to the fully skewed triclinic.
A crystal system (7 of them, about box shape) is not the same as a Bravais lattice (14 of them, box shape plus where extra points sit). Both describe lattice geometry, not chemistry.