Introduction
All of the genetics on this site assume that horses inherit their genes through Mendelian Inheritance. Mendelian Inheritance is punnet squares and the "capital letter" / "lowercase letter" alleles. Most of the information on this site is pulled from wikipedia, but I'll source anything I don't get from wikipedia.
Base Coat Color
The color of a horse's hair is determined by two pigments: pheomelanin, which is perceived as red to yellow color, and eumelanin, is perceived as brown to black. For simplicity, pheomelanin is usually called the "red pigment" and eumelanin is called the "black pigment". More can be seen on the "Base Coat Color" page.
Alleles
When we write a gene, we write it (usually) as the same letter twice. So agouti is AA, or Aa, or aa. This is because every horse carries two agouti alleles. These alleles aren't always the same; when they're different, we call the horse heterozygous. An allele is simply the variant form of a gene; most horses only have two alleles, or one, for each given gene.
Notation Conventions
Different genes are written differently, depending on author. Take the Cream gene, for example. A heterozygous cream gene can be written:
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nCr
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C / C
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Crcr
All are technically correct, but dA tends toward the first one for many reasons, the biggest being that it's the shortest and easiest to write. It doesn't require any superscript tags. This is only used with genes that typically have one allele, that all horses may not have a copy of. See, every horse has the agouti gene; that's why it's written AA/Aa/aa (depending on the alleles the horse has). But not every horse has a cream gene; there is no "off" allele for the cream gene. So it's typically written nCr to avoid confusion. Other genes treated the same:
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Cream: CrCr / nCr
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Pearlino: PrlPrl / nPrl
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Sabino: SbSb / nSb (Also written Sb1Sb1 / nSb1).
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Tobiano: ToTo / nTo
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Roan: RnRn / nRn
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Appaloosa: LpLp / nLp
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Champagne: ChCh / nCh
You can see the pattern. Many fantasy genes in dA follow this pattern as well.
Dominant vs Incomplete Dominant vs Recessive
A horse gene being dominant means that it needs only one allele to be expressed. Most horse genes are dominant; for example, having one copy of the A agouti allele is enough to turn a black horse into a bay horse.
A gene being Recessive means it needs two alleles to be expressed. The only recessive genes I'm aware of right now are Flaxen and Pearl; you need two copies of either to have any effect on the horse.
Incomplete Dominant is between those two; it behaves differently based on how many copies of the allele there are. Cream and Pearl are examples of incomplete dominant genes; CrCr looks different from nCr nprl looks different from prlprl.
The KIT Gene
Dominant White, Sabino, Tobiano, and Roan are all different alleles of the same gene, known as the "KIT gene". This means a horse can only have at most two heterozygous, or one homozygous.
Cr