Why symmetry matters
A crystal's external faces, growth angles, and cleavage directions are all consequences of how its atoms repeat in space. Two minerals with completely different chemistries can produce crystals that look almost identical because they share a crystal system. Once you can name the system at a glance, identification narrows from 5,000+ minerals to a handful.
The seven systems
Cubic (highest symmetry — fluorite, pyrite, galena, garnet). Hexagonal (six-fold axis — beryl, apatite). Trigonal (three-fold axis — quartz, calcite, tourmaline). Tetragonal (four-fold axis — cassiterite, rutile, scheelite). Orthorhombic (three perpendicular axes of unequal length — barite, topaz, sulfur). Monoclinic (one oblique angle — gypsum, azurite, orthoclase). Triclinic (lowest symmetry, three unequal oblique axes — plagioclase, kyanite, rhodonite).
How to use it in the field
Look first at the crystal habit — is it a cube? A hexagonal prism? A rhomb? Count the faces meeting at a corner. Check whether opposite faces are parallel. Within thirty seconds you should be able to rule out 4–5 systems and have your candidate list down to one or two. Combine with hardness and streak and you've solved most IDs.