A lattice (crystal lattice) refers to a specific, periodic configuration of atoms, molecules, or micelles which repeats throughout a structure. The basic space groups for crystal lattices are shown below.
There are also more complicated space groups such as the diamond lattice which is found in Silicon crystals or hexagonal lattices which are found for close-packing sphere structures.
Crystal structures typically split into multiple domains each having its own orientation. Crystals are typically categorized based on the typical size of these domains, which can be nanocrystalline, microcrystalline, single crystal, or amorphous (no crystal structure).
Miller indices are used to refer to the different crystal planes and directions in a crystal. For more information, see the Wikipedia page.