Shoes offer a vital boost in stats. Not all shoes are the best for showing though.
Western horses: Traction shoes. Discipline is worth TONNES in this event, and Traction shoes give the necessary boost. This is the ONLY shoe that does not give a stat bonus. However Western horses must have this shoe on in order to place decently.
Draft shoes: Gives +5 strength boost. Suitable for Jumping, Bronco, Driving, Dressage, 3DE, Foxhunters . . . anything where Strength needs to be highest.
Titanium shoes: Gives +5 speed boost. Necessary for Racers. Also suitable for Foxhunters, Gaited, 3DEers. . . probably some more, can't quite remember off the top of my head.
Padded shoes: +5 Endurance. Suitable for Endurance racers.
I believe some people also use Protective shoes (+5 movement, +10 surefoot I think) in some cases for Gaited for example. Whilst it does get a stat boost, it is also handicapped by the cap on Movement and surefoot. Some players think just having the shoe on gives a different boost, a minor boost. I don't know, I haven't tried

*edit*
Those shoes I've listed above are the most used shoes. Others I haven't listed aren't considered overly important or useful for eventing