Patterns that appear on bettas are the result of gene(s) that modify the way pigment appears on the fish. These patterns are not usually defined by colors, and multiple patterns can appear on a fish, making for a large range of possibilities.
Some patterns combined with specific color(s) might be given a commercial label (strain name/nickname) but they are not a single gene.
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Cambodian ( c )
The cambodian gene removes black pigment from fins and removes all pigment from the body.
Butterfly ( Vf )
Butterfly bettas show at least two bands of color around their fins, with the first band typically being the same color as the body.
Marble ( Mb )
I have discussed the marble gene in great detail here: What is a koi betta?
Mask ( Mk )
The mask gene causes the body color to completely (or mostly) cover the head, resulting in a solid colored fish.
Piebalds are similar to mask except they show no pigment anywhere on the head. Sometimes they are called "monkey face."
Dragon ( Dg )
A mutated metallic gene produced dragon, where the metallic scales appear thickened and silvery-white.
Grizzle is tied to the opaque trait and causes the pigment to appear sprinkled/splotchy over the body and fins.
Bicolor is a name for any betta with a body of one color and fins of another. Cambodians are not bicolor because they have no pigment on their body.
Cris-cross markings show when the blonde gene is crossed to an extended red/yellow fish, making black pigment appear from between the scales.
Extended red with royal blue iridescence creates the look of a purple bodied fish. This, paired with a white band around the fins, defines the salamander.
Dalmatian's show no definitive pigment in the body and only small spots of pigment in the fins. Red and orange is most common.
When the dragon scaling only appears on the head, the fish is deemed a monster.
Multicolor is the term used for any other non-solid betta that does not fall into another group.