The clear variety of quartz is called rock crystal or just quartz. It is more favored as a nice crystalline specimen or for cutting very large stones where size is the impressive feature. It is rarely found in jewelry today.
----------------------------
The purple variety of quartz is called amethyst. Amethyst runs in color from very light lilac (sometimes called rose de france) to the best deep purple with red flashes (sometimes called Siberian amethyst). More recently the color in amethyst is thought to come from the element iron in conjunction with aluminum (natural). Iron is not a purple color center, but when partially combined with aluminum and perhaps some form of irradiation, it does produce purple. Amethyst can be heated to produce citrine (yellow quartz), and the process can be reversed using X-rays.

Often when secondary crystal are present in the amethyst, they will bleach the color from those areas where they are present. Amethyst with lepidochrosite (FeO(OH)) crystals usually loses its colors from around the included area. This is probably due to a change in oxidation state near the impurity sites.

Another material containing amethyst is ametrine. Ametrine is a mixed color variety of clear quartz. It may have been created by zone heating in massive amethyst. It comes from limited localities, the majority of it comes from Bolivia. The mixed colors may have been created during the crystallization process with differing temperatures of formation even within a single crystal. It can be artificially created by zone heating amethyst.
-----------------------------
Citrine is the yellow to yellow orange variety of quartz. The lighter colors are sometimes called lemon citrine and the most sought after deep color is sometimes referred to as Madera citrine. It is named from the Madeira region of Brazil.

Please note: citrine is sometimes sold under the name of Madera Topaz, Spanish Topaz, or Smokey Topaz. This is unfortunate because it is a beautiful gemstone, but sometimes unscrupulous dealers will try to pass it off as the more expensive topaz. Most citrine found in today's jewelry began life as amethyst but was heated to between 470 - 560 F to turn it to citrine.

Yet another variety of citrine is called "oro verde" citrine. It has a light yellow color with slight greenish overtones. Rumor (maybe urban legend) has it that it started life in pile of discarded crystals that were too dark to be useful for anything by specimens. Anyway the story goes that a portion of this pile "accidently" made into a nearby fire and as luck would have it, it turned yellow-green. And thus was born oro-verde citrine. It is currently quite popular. Polished complete crystals often make it to market, and have dark phantom crystals imbedded in their bases.
-------------------------------
Smokey quartz, also called Cairngorm after its locale name in Scotland, is a brownish variety of quartz. It sometimes can be converted to citrine when heated. It is another of the varieties that is not found very often in jewelry,but very large stones are available and often cut. It to is sometimes sold as topaz in an attempt to raise its value. It is marketed under the name of Smokey Topaz.
--------------------------------
Rose Quartz, is pink quartz and is usually found in masses and not often a clean crystals. It is nearly always cloudy in appearance and thus not often seen in faceted stones. It sometimes contains titanium as micro-rutile crystals and these may produce a star pattern (asterism) in cabochons.

Clean crystals are normally only found in very small sizes, and are more suitable to collecting for themselves rather than for faceting. Most of the rose quartz makes it way to carvings or cabochons.
--------------------------------
Included quartz, this is quartz with secondary minerals trapped within the crystal structure. The minerals are often used as design elements in the stone cutting. One of the most common types of inclusion is rutile. Golden or silverish rutile crystals are often contain within quartz crystals and provide nice points of interest for the otherwise mundane rock crystal. Sometimes the rutile can be found associated with hematite. On rare occasion a hexagonal black hematite crystal can be found with golden rutile forming a 6 sided star in the hematite crystal directions, and this can be found as a quartz inclusion. They make spectacular gem stones and are highly sought after. (See example in another section.)

Tourmaline can also be found running through quartz as it is a well known associate mineral. Below are examples of black tourmaline (schrol) in a few faceted stones. Red/pink tourmaline and green tourmaline are also found in nature. The material is often called tourmalated quartz.

Lepidochrosite is an iron oxide-hydroxide that also may be found in quartz or amethyst. It produces bright red crystals that are easily discerned in the structure. Again several examples are shown below.

Cacoxenite is another mineral often found in quartz, especially amethyst. It is a complex hydrated iron-aluminum phosphate. It form golden needles that usually form parallel band in the crystal.