I know the American 'Bashkir' Curly Horse, which their are only about 4,000 in the world. Their are many theories to where they originated, but they did show up in herds of wild Mustangs. And many stories of people getting Curlies started out in a way such as this: they would have many horses, a few of them Curlies, and come some of the harshest winters would cause the death of many of the horses, and their Curlies would survive. (The Damele line of Curlies started this way. With the relization of this hardy breed they would catch more of them. But they crossbred a lot of them with more common breeds. But they did create some really good cow-horses.) Their are really interesting horses! I have one and I now know more about Curlies than any of the other breeds of horses I own.
It is interesting to find the stories of different lines of Curlies; Damele and Mead are some off the top of my head.
But the main thing that you will see that is clearly different with them is their hair. It is curly, thus their name. Most people who are allergic to regular horses find relief with the hair of the curly horse.
They are also said to have great personalities. While I might vouch for that, they are still individuals and they all are different. And the fact that their isn't as many of them gives that appeal. I have never met a mean curly, but I have only met 5 of them. They are gorgeous horses, and can be virtually any color. Seeing as they were crossbred with other breeds years ago sometimes you will find ones that look exactally like that breed just with curly hair. Then you will find some that are bred to more of the curly standard.