Definition of a Crystal

"A crystal is a body bounded by flat faces that is an external expression of an internal order."
From the Photographic Guide to Minerals of the World by Ole Johnsen

A mineral is a naturally occurring element or chemical compound, formed as a result of a geological process, and is usually crystalline. Most minerals are chemical compounds rather than containing just one element, and most, though not all, are inorganic. Synthetically produced stones are not regarded as minerals, as they are produced in a laboratory, and should always be referred to as synthetic, or expressed as such in another way. There are approximately 4000 minerals known on Earth, and new minerals are being discovered all the time, so don't expect to memorise every one of them! Most mineral names end in "ite" or "lite", which come from the Greek word lithos, meaning stone.
A crystal is a homogeneous body, meaning that it has the same chemical and physical properties throughout, and is formed by a regular lattice of atoms and molecules. Crystals form geometrically, and their outer shape will ideally be limited by flat surfaces, forming flat crystal faces. Crystals are crystalline, but the term crystalline can also refer to aggregates and other minerals that do not show a visible crystal structure.
A mineral is a crystalline solid, whether or not it is limited by plane faces. Some crystals are microscopic, and not visible to the naked eye; others have a crystalline structure that does not result in plane faces, but they are still regarded as crystalline, because they possess an ordered inner structure. As in the case of crystals, they are homogeneous, and are formed through a geological process.
Amorphous minerals are not crystalline at all, but are still of geological origin and are sufficiently homogeneous to demonstrate well-defined chemical and physical properties. The reason for lack of inner structure is usually due to rapid cooling, giving them no opportunity crystallise, for example obsidian (volcanic glass) and tektite (glass formed by a meteor strike).
Organics are generally considered non-mineral, and have formed through living things. Examples are amber (compacted tree resin), pearl, coral, coal and jet. Some crystal books will list these as amorphous.
The Crystal Lattice
In most solid substances, atoms and molecules move about freely with much empty space around them. Because crystals have a strong electromagnetic field, and are often formed under intense pressure, their atoms and molecules are closely packed in a strict order during growth. This forms what is known as the crystal lattice. The perfect symmetry of this structure will determine the outer shape of the crystal. 
Crystal Form and Habit
A combination of the chemical composition and inner structure of a mineral will determine its properties. Many minerals are related, either because they possess the same chemical compound, or are of the same type of crystal structure. The properties of a mineral will include the outer shape, hardness, cleavage, optical qualities, type of fracture and specific gravity. 
The term habit refers to the arrangement of faces preferred by a mineral. This can also refer to its type, and can include descriptions such as long, short, fibrous, needle-shaped, prismatic, equidimensional (containing crystals of roughly equal size), columnar, tabular, or compact. The term aggregate refers to an assemblage of crystals. An aggregate is considered to be granular if the crystals within it are equidimensional, and granular aggregates can further be described as fine, medium or course-grained according to the size of the crystals. Aggregates can be further described as scaly, hair-like, foliated, radiating, columnar, or wire-like, and also dense, massive, banded, stalactitic, botryoidal (shaped like a bunch of grapes), reniform (kidney-shaped), oolitic (small spheres), or pisolitic (slightly larger pea-sized spheres). In addition, some are described by the terms dendritic (moss-like appearance, although the root of the word comes from the Latin for tree), and arborescent (shaped like tree branches.) The distinction between crystal habit and shape is somewhat vague.
Properties of Minerals
Crystals and minerals, and particularly gemstones, are identified through several means. You will not be expected to know and understand more than hardness, but it may be useful to have an idea how the other properties are determined.
Hardness: The term hardness refers first to scratch hardness, and then to cutting resistance. Knowing the hardness of gemstones and minerals is of particular importance to the lapidarist or jeweller who will be working with the stone. This is usually referenced using Moh's Hardness Scale, which catagorises crystals from 1- 10, with 1 being the sofest and 10 the hardest.
Cleavage and Fracture: Many gemstones can be split along flat planes, known as cleavage. This is related to the lattice of the crystal. Fracture refers to the breaking of a stone causing irregular surfaces, and the type of fracture can sometimes be used to identify a mineral.
Density and Specific Gravity: Specific Gravity indicates the ratio of the weight of a specific material to the weight of the same volume of water. This has now been replaced by the term Density, which is expressed as grams per cubic centimetre. 
Colour of Streak: Because one variety of crystal may occur in many different colours, caused by impurities, colour itself is not considered to be diagnostic. Streak is determined by "streaking" the mineral on a rough porcelain plate. The streak left will be the inherent colour of the mineral, which is constant.
Refraction: Refraction occurs when a ray of light leaves one medium, such as air, and enters another, such as water or a crystal, and the interface between these two media. The effect is that of bending the light, such as when a stick is partially immersed in water. The amount of refraction in crystals is constant, and can be used in identification.
Dispersion: Dispersion occurs when a colourless crystal disperses white light into its spectral colours, in the way that a prism does. This is especially notable in diamonds, and referred to as "fire".
Absorption Spectra: This consists of the spectral colours of light as they emerge from a gemstone. Certain wavelengths, or colour bands, are absorbed, and the colour of the gem is formed from a mixture of the remaining parts of the original white light.
Transparency: The transparency or clarity of a gemstone is a factor in grading and evaluating. There are also a number of colour and light effects, too numerable to mention here, but worth having an awareness of.
Luminescence: The emission of visible light under the influence of specific light rays, most commonly ultraviolet.
Crystal Systems - The Seven Crystal Systems
Crystal shapes have been divided into seven groups or systems. There are also classes within each system, but for the sake of simplicity, we will not cover these in this course. Each system is determined by the crystal's axes (plural of axis), and the angles at which these axes intersect. As this involves a good knowledge of geometry, we will touch only briefly on this area of identification. Some books also look at the basic two-dimensional structure that determines the symmetry of each shape. This will also be included, as it is easier to understand and remember.  
Cubic System
Cubic System (also known as the isometric system)
All three axes are of equal length and intersect at right angles.
Based on a square inner structure
Crystal shapes include
Cube (diamond, fluorite, pyrite), octahedron (diamond, fluorite, magnetite), rhombic dodecahedron (garnet, lapis lazuli rarely crystallises), icosi-tetrahedron (pyrite, sphalerite), and hexacisochedron (pyrite).
Common Cubic Crystals:
Diamond, Fluorite, Garnet, Gold, Pyrite Silver, Spinel
Tetragonal System
Two axes are of equal length and are in the same plane, the main axis is either longer or shorter, and all three intersect at right angles.
Based on a rectangular inner structure
Crystal shapes include 
Four-sided prisms and pyramids, trapezohedrons and eight-sided and double pyramids.
Common Tetragonal Crystals:
Anatase, Apophyllite,Chalcopyrite, Rutile, Scapolite, Scheelite, Wulfenite, Zircon
Hexagonal System
Three out of the four axes are in one plane, of the same length, and intersect each other at angles of 60°. The fourth axis is of a different length and intersects the others at right angles.
Based on a hexagonal (6-sided) inner structure
Crystal shapes include hexagonal prisms (columns) and pyramids, twelve-sided pyramids and double pyramids.
Common Hexagonal Crystals: 
Apatite, Aquamarine, Beryl, Cancrinite, Emerald, Goshenite, Morganite, Sugilite, Zincite
Trigonal System (Rhombohedral System)
Axes and angles in this system are similar to the Hexagonal System, and the two systems are often combined as Hexagonal. In the cross-section of a Hexagonal crystal, there will be six sides. In the cross-section of a Trigonal crystal there will be three sides.
Based on a triangular inner structure
Crystal shapes include three-sided prisms or pyramids, rhombohedra, scalenohedra.
Common Trigonal Crystals:
Agate, Amethyst, Aventurine, Calcite, Carnelian, Chalcedony, Chrysoprase, Cinnabar, Citrine, Dioptase, Eudialyte, Hematite, Jasper, Magnesite, Phenakite, Quartz, Rhodochrosite, Rose Quartz (rarely crystallises), Ruby, Sapphire, Smithsonite, Smoky Quartz, Tiger's Eye, Tourmaline
Orthorhombic System (Rhombic System)
Three axes, all of different lengths, are at right angles to each other.
Based on a rhombic (diamond-shaped) inner structure
Crystal shapes include pinacoids, rhombic prisms, pyramids and double pyramids.
Common Orthorhombic Crystals:
Alexandrite, Andalusite (Chiastolite), Anhydrite, Aragonite, Baryte, Celestite, Cerussite, Chrysoberyl, Danburite, Dumortierite, Enstatite, Hemimorphite, Iolite, Peridot, Prehnite, Purpurite, Sulphur, Tanzanite, Thulite, Topaz, Variscite, Zoisite
Monoclinic System
There are three axes, each of different lengths. Two are at right angles to each other and the third is inclined.
Based on a parallelogram inner structure
Crystal forms include basal pinacoids and prisms with inclined end faces
Common Monoclinic Crystals:
Azurite, Brazilianite, Charoite, Chrysocolla, Crocoite, Diopside, Epidote, Gypsum, Hiddenite, Hornblende, Howlite, Jadeite, Kunzite, Lazulite, Moonstone, Muscovite (Mica), Nephrite, Neptunite, Orthoclase, Petalite, Serpentine, Sphene, Spodumene, Staurolite, Talc, Vivianite
Triclinic System
All three axes are of different lengths and inclined towards each other.
Based on a 'triclinic' inner structure, meaning 'three inclined angles'
Crystal forms are usually paired faces
Common Triclinic Crystals:
Amazonite, Amblygonite, Aventurine Feldspar, Kyanite, Labradorite, Pectolite (Larimar), Rhodonite, Turquoise, Ulexite
Amorphous System
No crystal structure. Most of these are either cooled too quickly to crystallise, (such as obsidian or moldavite), or are organic (such as amber).
Common Amorphous Minerals:, Amber (not considered a mineral in most systems), Moldavite and other Tektites, Obsidian, Opal