The horse with a roman nose has a convex profile. Convex heads are generally associated with Draft horses, Baroque horse breeds and horses from cold regions.
Roman noses are common (and sometimes are considered a breed trait) in Iberian/related breeds, such as Lusitanos, Andalusians, Lipizzanners, etc. Many of the heavier Appaloosa horses have them. They are frequently seen in Standardbreds.
Some “exotic breeds” such as the Barb often have the roman nose trait. However, a roman nose can be found in just about any horse breed. They are probably least likely in Arabians because breeders often strive for that “dish-face” or slightly concave profile.
Roman noses are more commonly seen on the heavier breeds of horses that are used primarily for pulling. Riding horses usually sport the more-dished face. The dish allows the horse to see better in front of itself, making it slightly more suitable for riding.
Roman noses are not considered a fault, unless the specific breed is not meant to have one! For example, if an arab had a roman nose, that would probably be considered unattractive to aficionados of that breed.
Genetically, this trait likely plays a role in warming air as it is inhaled, but may also influence aerobic capacity.This is also true in human noses. People from colder climates usually have larger noses than people from temperate climates, too.
Compare it to the smaller refined nose of the Arabian, which originated in hot desert conditions.
Local lore says a horse with a roman nose will be more willful and stubborn than most horses. In India, a horse with a pink roman nose and light colored eyes is very desirable.
Many roman nosed horses have extra large heads in proportion to their bodies and big almost floppy ears. Some people argue that the Roman faced horses are better adapted, they have larger teeth and more room in the mouth for grinding as opposed to the tiny faces we have bred into a lot of other breeds.