Pigments that appear on bettas exist in the skin as chromatophores, which are cells that contain pigment. They are best described as existing in structured layers. In betta splendens, iridescent is on top, with red on the bottom. Cellophane is the true bottom layer, but it holds no pigment, and is often overlooked in color layer charts.

• The iridescent layer holds royal blue (⬤Bl), steel blue (⬤Bls), turquoise (⬤Blt), and iridescent (❂Si) which is not a true pigment, rather crystallized guanines that reflect light and appear as bright colors or white. It also holds metallic (❂Mt) and opaque (◒Op), which changes the depth of the guanines within the layer to be lower, and therefore less shiny.
• The black layer holds melano (black) (⬤/⬤bm), black lace (a second black gene, slightly lighter then melano) (⬤blc), and red-loss (◎Rlos), which is not a color itself, but causes a fish to lose red pigment as it ages.
• The red layer holds red (⬤R), non-red 2 (orange) (⬤nr2), non-red 1 (yellow) (⬤nr1), and blonde (◌bd), which is not a color itself, but responsible for reducing the density of black pigment, which results in a "washed-out" color. The layer also holds the extended red (◉Er) gene, which is responsible for increasing the density of the red pigment.
• The cellophane layer holds no pigment, making the fish appear white from a distance, but slightly pink up close. (Kind of like how you can see the blue and red veins through the skin on your arms and hands.)

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• Basic Colors
• Combined Colors & Modifiers
• More Science
  

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Click images for original sources.

Royal Blue

Steel Blue

Turquoise

Turquoise (green)

Red

Extended Red (allele = Er )

Orange

Yellow

Melano

Brown (unoxidized melano)

Black Lace

Cellophane

There are many different genetic routes you can take to reach a specific color. For example, a "pink" betta could be as simple as a red crossed with an opaque white, resulting in pastel red or "pink" offspring. Alternatively, a red betta with the blonde trait crossed with a steel blue or turquoise could produce a "pink" betta. This is why it's important to know the genetic background of the betta before you decide to breed, because a [Pastel-Red] x [Blonde-Red x Steel Blue] is going to give you [Pastel-Blonde-Red x Steel Blue] which could show as a very pale purple, a multicolored washed out red/blue, and a multitude of other options.

Click images for original sources.

Super Black (melano x black lace)

Cornflower Blue (steel blue x turquoise)

Purple (royal blue x red x steel blue)

Green (yellow x turquoise x wild type)

Opaque ( allele = Op ) Causes white, powdery-pale appearance over body and fins. Density of Op changes how the colors appear on the fish, with high density producing white, and low density producing pastels.

Pink ( red + Op)

Pastel Blue (royal blue + Op)

Bicolored Pastel

Metallic (allele = Mt ) Changes the structure of the crystallized guanines in the iridescent layer, causing a metallic sheen.

Copper (unoxidized melano + metallic)

Gold (yellow + metallic)

Silver (Black + bd + metallic)

Spread Iridescence (allele = Si ) Causes iridescence to spread over the body and fins.

Normal Iridescent (allele = si+ ) & Non-Metallic (allele = mt+ ) (Wild Type alleles = + )

Blonde (allele = bd ) Reduces the density of black pigment, causing a washed out color. (Pictured is a blonde red.)

Chromatophores are divided into groups:

• Xanthophores and erythrophores both hold pteridines and carotenoids, or yellow and red/orange pigment respectively.
  
• Melanophores holds melanin, which appears black or dark-brown.
• Iridophores reflect light using crystallized guanine.
• Microscopic pteridines have fine, reflective surfaces, giving them the appearance of blue and green hues thanks to light. This is called structural coloration and is responsible for the metallic sheen.

In short...

The iridescent layer holds iridophores & pteridines.
The black layer holds melanin.
The red layer holds xanthophores & erythrophores.

All bettas, even solid "pure" colors, have these chromatophores.
Their densities and their enzymatic structures are what alter the colors we see on the fish.