This post has been revised with new photographs here.
The base colors of horses are bay, black and chestnut, possibly with the addition of wild bay and seal brown (tan-point.) These colors are distinguished by where red and black pigment are found, both where on the whole horse and where on individual hairs. I’m going to go into more detail this time on what determines these base colors.
Red pigment in horses (more correctly, phaeomelanin) can appear brownish red to copper, sometimes approaching gold, in the absence of dilution factors. With dilution factors, it can include white, cream, tan, yellow and gold shades.
Black pigment (more correctly, eumelanin) is black in the absence of modifying genes. In horses, the genes that dilute black to blue-gray or black to chocolate brown are not known to occur, though they do occur in other species. Some dilution genes in horses do affect black, changing it to shades from sepia to dirty white or even pure white.
Black. This mare actually has a few brown hairs, but this is not unusual for blacks in summer coat–black is subject to “sunburning” of the color.
The Agouti locus is known in almost all mammals. It codes for a protein that affects more than coat color, and is complex to sequence. In general, however, more red pigment is dominant to more black pigment.
The Agouti locus is given the symbol A. Agouti alleles are A with a superscript showing the particular form of the allele. Thus Aa is the symbol for recessive black, also called non-agouti. At stands for seal brown (black with tan on the inner legs, flanks and muzzle, very hard to tell from black with the mealy gene) which is also called tan-point in some mammals. AA is the symbol for bay. A+ is the so-called wild bay, where some red pigment appears on the lower legs.
Every horse has two alleles at each locus. If one allele is dominant to the other at the same locus, that is the allele that determines the color of the horse—if the extension locus allows it to. The order of dominance at the agouti locus is wild bay is dominant to all others, bay is dominant to black and tan-point but recessive to wild bay, tan-point is dominant to black but recessive to both bays, and black is recessive to the other three alleles. This means that two recessive blacks can produce only black foals, while two wild bays can produce any color.
The agouti gene, by the way, was named for a South American rodent, the agouti. It was originally defined as controlling banded hair, seen in many wild animals. In fact, banded hair (black tips on red hairs) can be found on most bay horses, though you’ll need a magnifying glass and very good light to find it.
Chestnut paint. This particular horse appears to have the frame allele, judging from the face and lower legs.
The Extension locus is given the symbol E. The wild-type allele, E+, allows the agouti alleles to be expressed. There is also a recessive allele, Ee, which suppresses the black pigment. Not completely—a horse with two Ee alleles can still have black whiskers and may have black hairs scattered throughout the coat. But it will not have the black mane, tail and lower legs of a bay. In fact, an EeEe horse will be a chestnut, regardless of what may be at the Agouti locus.
E may also have two alleles dominant to the wild-type allele. These are dominant black ED and countershading, EB. At the E locus, alleles with more black are dominant to alleles with more red. Further, the E locus can hide what is present at the A locus. An ED horse will be black regardless of what alleles are present at the A locus, and an EeEe horse will be chestnut regardless of what is present at the A locus. The word epistatic is sometimes used to define this relationship between loci—Extension is epistatic to Agouti.
Note that I am following Sponenberg, Equine Color Genetics Third Edition, plus my own observations on hair color.