Horse is a non-supply pile. There are 30 of them. You get one via a card saying "gain a Horse" or something. If the pile is empty you can't gain one, but let's hope that doesn't happen much. Horse looks like this:
So Horse is a one-use Laboratory; it puts you up a card and then is gone. Or, you know, you don't have to play it; you can Remodel it or something. It's there for you. Once in your deck, it's just like Experiment, from Renaissance; it turns out, they were experimenting on Horses. Eight total kingdom cards use Horses, and four landscapes. Here are some of those kingdom cards.
Supplies is an easy one: when you play it, you get a Horse. The Horse goes onto your deck, so normally you'll just start with it next turn.
Scrap gives you a choice of bonuses, with more stuff for trashing a more expensive card. You could for example trash an Estate and get +1 Card and +1 Action, or trash a Horse and get +1 Buy, a Silver, and a Horse.
Livery makes everything that's expensive enough come with a Horse that turn. This works on cards you buy, or cards you get other ways. With two Liveries, everything comes with two Horses. With two Liveries, and you buy Farmland, trashing Remodel to gain a Border Village which comes with a Laboratory, you gain six Horses. Many Horses could be yours.