Horseshoes can be crafted using steel, bronze, mese and diamond (4x ingots - 2 down either side with 1x block top middle) and placed on a horse by right clicking with the item. These can make horses run faster or jump higher while riding depending on tier.
When riding a horse monsters will generally attack the horse first to get to player riding it, when horse dies the player is dismounted and it will drop any shoes or saddles in use as well as some horse meat.