What is a "taffy" horse?

"Taffy" is a term more commonly used in Australia to describe a horse that carries the silver dilute gene.
The silver gene dilutes black pigment to a lighter color, it's almost like the opposite of the cream gene (which is the gene that turns chestnuts into palominos or bays into buckskins, diluting red pigment).

The silver gene, on a black horse, will often create a chocolatey brown body with a flaxen mane and tail. On a bay, it will lighten the legs to chocolate and make the mane flaxen, but the body will stay the same (red). On a buckskin, it often makes the horse's mane flaxen, leaves the body cream, and diltues the legs to a chocolate color.
It has no affect on a chestnut horse.

However, just because a horse is a brown or chocolate with a lighter mane/tail doesn't mean it carries the silver gene. Some chocolate palominos may resemble silvers but aren't.
You could tell if your horse was a palomino or silver either by doing a genetic test (a red factor test, which tells if it's a red-based or black-based horse), or you might be able to tell if you knew the color of its parents.

Some of this genetics terminology might be confusing, there's a site with photos, examples, and explanations on different horse colors at the Horse Color Genetics section.

So, the term "taffy" is mainly used in Australia to describe this type of horse. Other places call it "silver", but I've also heard the color called "chocolate" or "flax" (especially in the Rocky Mountain Horse breed).