What is a horse beauty contest called?

The general term is 'horse show'. The specific class is refered to as a 'halter class'. This class is judged on how attractive and correct your horse is. Halter classes differ from other 'in-hand' classes such as showmanship which is an in-hand class but is judged on how well the handler grooms and shows his horse.

There is a lot to know about showing in halter classes so if you are interested go online. There a lots of sites to educate yourself and give you an idea if your horse is suited for this.