Hi, some domestic horses don't wear shoes, especially those kept in fields....well they may have their front hooves shod, but not the back as then they can do serious damage to other horses when fighting. It's mainly to protect their hooves and hard surfaces; roads, stable floor and stable yards. Well that's my humble opinion....but I haven't ridden for ?? years....... :o[
Even domestic horses which are not shod still need their feet trimming and filing down about every two months. Wild horses cover a lot of rough ground searching for food and therefore wear their hooves down naturally. Shoes are fitted to protect the feet from hard surfaces such as roads and can also help hoof and foot medical problems.
Its realy a matter of personal opinion (and finance of course) as to how a horse is shod but a horse that has no shoes or just shoes on the front usually means they dont do as much road work as a horse that is fully shod