Wry nose is technically called "deviated rostral maxilla and associated nasal septal deviation."
In layman's terms - it's a crooked muzzle. The upper jaw and nose is turned to one side, while the lower jaw remains straight. This results in breathing difficulties and misalignment of the teeth. Severely affected foals may be unable to nurse from the source and may have to be bottle fed.
It appears not to be genetic, and is suspected to be caused by poor position in the uterus. The first sign is often a difficult birth - the twisted nose can be severe enough to get caught in the birth canal. Some foals with wry nose also have other deformities, including cleft palate.
Mild cases often don't need treatment. Severe wry nose is treated with surgery in modern times, but it's expensive and doesn't always result in an animal that has normal appearance and function. Some foals with a severe deformity can die because they can't breathe well enough. However, when the surgery is successful you sometimes can't even tell there was ever a problem.