Types of Gemstones: Kinds of Stones People Value
To outline types of gemstones, let’s remember what a gemstone is. Gemstones, also called gems or jewels, are usually defined as pieces of some mineral used for making jewelry, after being carved or cut and finally polished.
The ancient (though still used) classification of gemstones was limited to dividing gems into precious or semi-precious.
So, can we simply say that jewel is a piece of a certain mineral used for making jewelry? Yes and no – because some of the jewels are not minerals.  For example, pearls, amber, coral, or jet coal (known as black amber or gagate) are of organic origin. Artificial and synthetic gemstones are actively used in jewelry, despite being created not by nature but by human workmanship. 
The very word ‘precious’ implies that the common characteristic of these stones is their value. So, division of  gemstones into precious-and semi-precious is based on such subjective criteria as their aesthetic value and rarity (common things are rarely considered valuable, aren’t they?)
At present, only four gemstones are considered precious: diamond, emerald, sapphire, and ruby. All other gemstones are called semi-precious stones.
Gemstones from Chemists’ Point of View
Of course, jewelers and gemologists have more precise classifications to describe gems. They divide gemstones into groups, species, and varieties. Chemical composition plays important role in modern classification of gemstones.
For example, we all know that diamond consists of carbon atoms (C).  Black amber also consists mostly of carbon; it is the crystalline grid that makes all the difference.
For a gemologist, Beryl is a ‘mineral  species’, with different ‘varieties’ including emerald, aquamarine, red beryl, goshenite, heliodor, morganite.  They all fall into Beryl Group and have the same chemical elemental composition  - Al2Be3Si6O18, though emerald is green, aquamarine is blue, red beryl is (predictably) red, heliodor is yellow, morganite is pink, and goshenite is colorless.
Rubies’ crystalline grid is built of aluminum oxide molecules (Al2O3). Ruby is a red variety of corundum (which is a mineral species), and sapphire is variety of corundum of any other color. Some kinds of sapphires have their own names, for example, padparadscha is pinkish orange, Yogo sapphire is colored from blue to purple.
One and the same stone can come in a great variety of colors. For example, tourmaline is known to be almost any color; but, in fact, tourmaline is not a gem but a mineral species. Tourmaline Group has many varieties of different color but of the same  chemical elemental composition  - (Na, Ca)(Mg,Li,Al,Fe2+)3Al6B3Si6(OH) Here are just some of them:
Schorl  is black, achroite  is colorless, canary tourmaline is bright yellow; chrome tourmaline is  deep green,  dravite  is brown.  Elbaite  can be either green or multicolored. Indicolite is blue. Paraiba Tourmaline  is neon blue. Uvite comes in various shades of green and brown, brownish-yellow, red, reddish brown, purple, gray, white and black.  Cat's eye tourmaline  displays a cat's eye effect, so don’t confuse it with variety of Quartz with the same name.
Different colorings of the same gemstone is caused by presence of impurities, or extraneous substances.
Let’s take a look at some of the stones, which belong to Quartz Group and all consist mostly of SiO2.
Purple coloring of an Amethyst is caused by traces of iron or manganese.
Citrine is colored yellow due to presence of iron.
Smoky quartz is gray or brownish due to extra aluminum.  
Rose quartz is tinted by titanium.
Natural aventurine, or flamboyant quartz can be of different colors, from various shades of green to creamy white, orange and gray. Due to mineral inclusions, it seems that flickers are sparkling from inside the stone. This lustrous effect is called aventurescence.
There are a lot more semi-precious stones actively used in jewelry, which are actually varieties of quartz:  sapphire quartz (an opaque indigo-blue stone). tiger's-eye, hawk's-eye, cat's quartz, and many more.