Natural pigments come from a wild array of sources from pregnant beetles to precious stones. When used as paint they offer a natural palette of harmonious colours , especially in comparison to the synthetic look of chemical pigments. Natural pigments are your oxides, cadmiums, carbons, ochres, and siennas. See a longer list of natural pigment names at the bottom of this article. The above image shows the pigment used to make Ultramarine Blue. It's a mineral source mined from the earth, a semi-precious stone called Lapis Lazuli that is ground to create the pigment. 

Natural pigments can be inexpensive when created from inexpensive sources - like carbon black made from charred wood. Some are more expensive due to the value of the raw material and/or the process it takes to produce the pigment. Cadmium red is made by baking cadmium metals which is an expensive process. Another point of interest is that some pigments are heated to change the depth of colour. Cadmium used in the creation of cadmium red deep is baked longer than cadmium red medium and cadmium red light ... well you get the picture.

When looking to uncover if your pigment is synthetic or natural you will come across the confusing terms organic and inorganic. The confusing part is that organic pigments refers to synthetic/chemical pigments. Counterintuitive right?

So:

inorganic = natural pigments

organic =  synthetic/chemical pigments

Below you can see the natural "greying down" of natural pigment Ultramarine Blue in comparison to what happens below with Phthalo Turquoise. When you add white to the synthetic/chemical colour Phthalo Turquoise it stays very intense or "high key". This is a key difference between natural and synthetic colours.

 ULTRAMARINE BLUE:

PHTHALO TURQUOISE:

Here's a list of common natural pigments ( I may have missed some):

Ultramarine violet: (PV15)
Cobalt Violet: (PV14)
Manganese violet: (PV16)
Ultramarine (PB29)

Cobalt Blue (PB28)

Cerulean Blue (PB35)

Prussian Blue (PB27)

Viridian (Cr2O3)

Chrome green (PG17)
Viridian (PG18)

Cadmium Yellow (PY37)
Chrome Yellow (PY34)
Aureolin(also called Cobalt Yellow) (PY40)
Yellow Ochre (PY43)
Naples Yellow (PY41)
Titanium Yellow (PY53)

Cadmium Orange (PO20)
Chrome Orange:  (PbCrO4 + PbO)

Cadmium Red (PR108)
Sanguine, Caput Mortuum, Venetian Red, Oxide Red (PR102)
Burnt Sienna, Raw Sienna (PBr7)
Vermilion (PR106)

Raw Umber Burnt Umber  (PBr7)
Raw Sienna (PBr7)

Carbon Black (PBk7)
Ivory Black (PBk9)
Vine Black (PBk8)
Lamp Black (PBk6)
Iron black (PBk11) 

Titanium White (PW6)
Zinc White (PW4)

Image source [1]

Image source [2]

Image source [3]