I think this is a perfectly valid question, but only someone competent with firearms should do this. There are other safety considerations as well, such as the nature of the gun, obstacles in the area (rocks, concrete, glass), what else is behind the target, and so on. So without going into a long lesson on safe firearms handling, the only place to shoot a horse with immediate fatality is in the brain. Unfortunately, the brain isn't actually all that large so if you miss, you can cause suffering.
From the front of the horse, the best place to aim for in in the centre of the head about 2/3rds of the way up between the eyes and the ears. Not straight in either, but at a slight angle -- so slightly less than 90% with the smaller angle being on the down side (larger angle toward the ears). Sounds confusing... sorry.
Another option is from behind. Imagine the horse grazing, the location is just behind the ears and aim slightly forward.
Either way is so quick you don't have time to even think of anything. You just decide it needs to be done, squeeze and it's over.
Use a gun with a magazine so that if something does go wrong, you can resolve the issue immediately. I've heard that a .22 will do the job, but I won't use less than a 7.62 for a large animal. I've used a .22 on a goat and other smaller creatures, but not a horse. Reason: a bigger hole does a better job. Plus if you are in an emergency situation and need to resort to a less humane shot (lungs), you can still get it done. Sad, but true. Always use soft point ammunition; again, bigger hole.
I don't use shotguns much, but I would imagine anything with a slug over 16 gauge would work. Though I'd probably go 12 g. I prefer the rifles for the magazine option. And I just like rifles better overall. I guess the wussie girlie part of me shines through on that. Plus rifles allow me to shoot bears from much longer (read "safer") distances!
There are other emergency ways to kill an animal if needed also. Some are pretty graphic, but still effective. My husband once got in a tangle with a dying deer (he thought it was dead -- ALWAYS CHECK!). Anyway, he had his axe with him, so he severed the jugular vein and the deer bled out quickly. This takes some muscle power and no hesitation. Don't underestimate how hard it is to cut through fur-covered skin, sinew (or whatever it is) and the vein itself.
There are very few animals you can kill by hitting them on the head. Don't try it. You are more likely to end up with the wrath of the animal on you instead.
That's my sad and sober input for today. Necessary information when you live away from immediate help though. I've known how to kill an animal since I was about 12.