There are two genes that cause a wiry or curly coat in horses. (And, like poodles and some other curly haired animals, they're hypoallergenic).
The first of the two genes is dominant. In homozygous form, as discussed under hairless horses, it often results in a thin mane and tail that may shed out altogether in the summer.
The second is recessive and shows up primarily in the Missouri Foxtrotter breed, resulting in an unexpected curly coat (recessive genes can hide for generations).
Curly horses are more common in America, where they have been bred into a specific breed, the American Bashkir Curly (Which has no connection to the Russian Bashkir horse - the name comes from a mythical breed connection that has since been disproved). It's also bred to have a quiet temperament and be easy to handle and an easy keeper. The American Bashkir Curly does not have a specific, settled type, and the registry will accept anything with a curly coat, but the actual ABC type is a cobby animal in the 14 to 15 hand range.
Being a mutation, the curly gene occasionally shows up in other breeds.
This picture of an American Bashkir Curly clearly shows the "rough" looking coat, curly (almost dreadlocked) mane and curly hair inside the ears. (Source: Penella22 via Wikimedia Commons).