Amorphous silica is noncrystallline and deposited as a gel with water content.  The most obvious example is opal, which comes in many varieties.  Common opal, which is pale colored, precious opal with its play of color, and fire opal which is red or yellow, also with a play of color.  Opal wood is petrified wood with opal as the petrifying material.  Tripolite is opalized diatoms (small critters) which forms a powder used for polishing, and hyalite which is opal crusts on rocks.

Another example of amorphous silica is silica glass, formed when lightning strikes quartz (fulgarites) , or when quartz is struck by a meteorite.  It's quite rare  because the temperature required to melt quartz is quite high. 

Gem varieties of silica are opal and quartz crystals in various colors:  rock crystal, rose quartz, amethyst, smoky quartz, milky quartz and citrine.  Sagenitic quartz has inclusions such as rutiles (rutilated quartz) and quartz pseudomorphs are minerals in which the structure has been replaced by quartz.  An example is tigereye, in which the original asbestos fibers  are visible.

 Cristabolite is  the form of silica that remains stable at high temperatures.  Quartz turns to cristabolite at 1055 C and the change is permanent. Cristabolite is often seen in obsidian as the "snowflaek" pattern.  Similar to cristabolite is tridymite, also found in high temperature minerals.