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Bay horses are identified by having a reddish brown body and black points (mane, tail, lower legs). The bay color is caused by genes at two different loci, A and E. The presence of the E gene codes for black color, and the A gene restricts the black color to the points. So bay horses are called A-E-.
Black horses have no A gene; they are aa at the A locus. Their genotype is aaE-. Lack of the A gene allows the black color to "spread" over the entire body; the presence of the A gene restricts the black color to the points.
Chestnut horses can carry the A gene, but since they have no black pigment (ee), they will not display evidence of the A gene if they carry it. An aaee horse and an A-ee horse look the same; both are simply chestnut. An A-ee chestnut can, if bred to a black (aaE-), produce a bay foal (A-E-), if the black parent passes on one of its E genes.
Bay comes in a lot of shades.
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Mahogany bay |
Light bay |
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Blood bay |
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Bay foals usually have cream-colored hair on their legs. When the foal coat sheds out, the black color of the legs is revealed.
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The blond baby fuzz on the legs of the two foals above makes them resemble chestnuts, but chestnuts never have black manes and tails, as these two have. The bay foal below has legs that are a bit blacker.

What is brown?
Breed registries, being political organizations, enjoy defining their own terms. Some registries, such as the Jockey Club, which registers Thoroughbreds, have a color they actually call "dark bay or brown." This ambiguously refers to any dark but nonblack horse with black points. A really dark A-E-, in other words. As I mentioned in the black section, true chocolate brown with no reddish cast is extremely rare in the horse. I do not have a photo of a horse this color. But when horsepeople refer to a horse as "brown," they usually mean one that looks like this:

This horse has a dark brown body and black points. I call this color dark bay, but many people call it brown, and you know what? That's OK. Your mileage may vary.
Here are more "dark bay or brown" horses.
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The horse at left is a shade often called seal brown. These are the darkest of dark bays (or browns, if you prefer), and also feature light brown patches on the flank (easily noticeable in this horse), behind the hip, behind the elbow, inside the upper legs, and around the muzzle and eye.