The "black" gene is actually called "Extension of Black" and is represented with an "E" or "e".
Now, the first thing you have to know, is that all horses produce red pigment. Even the black ones! What the "black gene" does is add an additional pigment into the hair shaft that is darker then the red, and thus is all you can see with your eyes. To wrap your mind around this, think about mixing paints. If you take bright red paint, and start adding in black paint, the red gets darker. Add in enough, and it looks black, or so close as to make no difference.
When people refer to "non fading blacks" what they are talking about is a horse with so much black pigment in its hair that even sun fading doesn't turn it brown (i.e. let that red pigment become visible). How dark a horse appears is controlled by one of those genes that we don't fully understand. I'll call it the shade or hue. This phenomenon is easily visible in chestnut/sorrel horses. It's what makes the difference between liver chestnut and golden chestnut, and why we use the terms chestnut and sorrel to describe horses that are genetically identical at their extension loci.
Think of the shade this way: If you add a single drop of red food coloring to white frosting, you get pink. Add more red, and the pink gets darker. Add enough, and it's burgundy. There isn't much science on this gene, but we can extrapolate its existence from looking at the horses around us. I mention this because when talking about black and red, questions on the shade always come up.
Now, this is a gelding, but if we forget about that, and look at the "what ifs" we can talk about what genes he could pass on. Rover would never be responsible for a black foal. He can only pass on a recessive copy of Extension (e) to any offspring. That's all he has to give!
If bred to a mare with a dominant Extension allele (E) then the resulting foal could be black, but it would be from the mare's genetics, and not good ol' Rover here. This is a good point to keep in mind that no foal is all the fault of a single parent. Stallions get this all the time. Owners of homozygous black stallions stating that their horse will have nothing but black babies. If the mare adds bay, or grey, or roan, or dun, or creme, or any combination of genes, the resulting foal won't be "black" . Yes, it will still have at least one dominant copy of Extension, but that doesn't mean it will LOOK black.
See, extension is the foundation of all other colors. Here are a few horses that have at least one dominant Extension allele. Often times I will refer to this type of horse as "black based".
So, as you can see, having a copy of the black gene does not make a horse black. It CAN, but it doesn't always.
Examples of red based variations are colors like palomino, grey, gold champagne, liver chestnut (which can appear black) and red dun. I'm sure there are more. In my opinion, modifying the red pigment through other genes doesn't seem to make as drastic of a change though, as modifying the black pigment through additional genes.
Now, a common problem that people have, is that they think that because there are more of something, it means its dominant. And just because a horse doesn't LOOK black, doesn't mean it isn't black based. Look at Amber, the first horse listed again. She's all shades of silvers and golds, but she's still black based! So the prevalence of horses that are "red looking" such as bays, chestnuts and sorrels doesn't change the fact that a single dominant Extension allele will always show its effect in some manner.
The other genes involved in horse color also play a role in what color a horse is. Just having a dominant extension allele isn't enough. Even though many of us want to, we can't simply look at ONE gene, and breed for that. It's no different then breeding just for, say, leopard horses and forgetting all about things like conformation and temperament. Breeding horses requires a lot of factors to be taken into consideration at the same time.
Next up... Agouti, the Bay gene.
A Note about Sugarbush Draft Horses
I see it over and over again, and no matter how many times it's said, it's still wrong. "Sugarbush Drafts are just an Appaloosa Draft Cross". Uh.... no. The Sugarbush Draft Horse was a breed created many years ago in Ohio. While the initial cross was made using Percherons to Appaloosas, in the many generations following, the breed has been solidified into a consistent type. Saying these horses are "just" a draft cross makes as much sense as saying that AQHA horses are "just" a Thoroughbred cross, American Cream Drafts are "just" a dilute Belgian, or that Morgans are "just" a grade.