lattice: an imaginary pattern of points (or nodes) in which every point has an environment that is identical to that of any other point in the pattern
2-D point groups: the 10 possible combinations of a rotation axis and mirror line (plane) in two dimensions; these 10 planar point groups are analogous to the 32 3-D point groups
3-D point groups: the 32 possible unique combinations of rotation, reflection, inversion, and rotoinversion symmetry elements in three dimensions; these 32 combinations of symmetry elements are also known as the 32 crystal classes; the morphology of every crystal must be based on one of these collections of symmetry
crystal classes: see 3-D point groups
crystal systems: the 6 (or 7) groupings of the 32 crystal classes done based on the presence of common symmetry elements: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal (rhombohedral), isometric
2-D plane lattices: the 5 lattices that represent the only ways to periodically repeat points in two dimensions: oblique, rectangular, centered rectangular, hexagonal, square
3-D space lattices: the 14 lattices (Bravais lattices) that represent the only ways to periodically repeat points in three dimensions
lattice types: primitive (P), body-centered (I), face-centered (F), rhombohedral (R) or end-centered (A,B,C)
2-D plane groups: the 17 unique combinations of the 5 plane lattices and the 10 planar point groups. Some of these plane groups have glide line (plane) symmetry elements representing the combination of translation with reflection.
space groups: the 230 unique combinations of the 14 space lattices and the 32 3-D point groups. In addition to glide planes representing the combination of reflection and translation, some of the space groups have one or more screw axes representing the combination of rotation and translation. You can reduce the space group notation to one of the 32 translation-free point groups by discarding the lattice type designation and replacing glide planes by mirrors and screw axes by their isogonal rotation axes.
symmetry elements without translation: rotation axes, mirror lines or planes, center of symmetry, rotoinversion axes
symmetry elements with translation: glide lines or planes and screw axes
crystallographic axes: conventionally chosen axes with specific length and angular relations that provide (literally) a framework for us to use to describe both the internal symmetry and the external forms of crystalline materials
Miller indices: the conventionally agreed upon alternative to the system of face intercepts as a short-hand for relating the orientation of a crystal face to the crystallographic axes. Remember both the process (relative intercepts, invert, clear fractions) and the conventions for reporting the result; (100) is a single face while {100} is the collection of faces symmetrically equivalent to the face (100). The number in braces {100} makes up a 'form'.
Forms: a group of crystal faces all of which have the same relation to the elements of symmetry. Forms can be either closed (completely enclosing space) or open (they would not hold water in all orientations). The same Miller index is possible in every crystal class (all could have a (100) face) but the number of faces that belongs to a form is determined by the symmetry of the crystal class.