My horse won't take anything that isn't "horse food." He loves apples, carrots, (most) commercial horse treats, alfalfa cubes, etc. But he doesn't like peppermints, sugar cubes, potato/corn/banana chips, bread, etc. I tend to offer him bits of whatever I'm eating now just to see if he'll take it, and he always sniffs it and turns away without even tasting it.
Most fruits/vegetables are OK for horses, but a few are not (broccoli, cabbage, potatoes, onions, etc.) so it's probably best to do some research before grabbing whatever is going unused in your crisper drawer and giving it to your horse
Most things are probably OK in small quantities but can be problematic in larger quantities. Offering a bit of bread from your sandwich is fine, but you wouldn't want your horse getting at a whole loaf of it. I'd stay away from meat and dairy products, though.