The carousel was originally built by The Philadelphia Toboggan Company, one of the most prestigious carousel companies, in 1917. In 1967, Walt Disney purchased the handmade carousel (and horses) from the Olympic Park in Irvington, New Jersey. Arrow Development, in which Disney then owned a 1/3 interest, produced new engineering drawings for the mechanism and horses. The original name of the carousel was Liberty, and the maidens can still be seen on the top of the carousel. Every horse is different. 
There are conflicting stories regarding whether one of the horses on the carrousel is "Cinderella's Horse". The horse in question is in the second rank of horses, and is the only one that has a golden bow on its tail.
Cast Members refer to this horse as one that belongs to Cinderella and it has been referred to as such in various Disney publications. Cinderella has been depicted as riding a horse in various pieces of collectable sculpture and artwork, such as a 2001 Limited Edition lithograph.
However, in an interview with Isle Voght, a park employee responsible for restoring the ride along with John Hench, she gives her own reasons for doubting the veracity of the Cinderella's Horse story, namely that the princess would not have a horse on an inner row, and that her horse would be decorated much more elaborately than the others. Also, she states that Cinderella never rode a horse in her film.
The Disney Imagineers did not include the story of Cinderella's horse in the 2010 rewriting of the carrousel backstory.
On June 1, 2010 the name was changed from Cinderella's Golden Carrousel to Prince Charming Regal Carrousel. The name change helps tell the rest of the Cinderella story that inspired the carrousel. Here’s the official story behind it:
Following their fairy-tale romance and happily-ever-after wedding, Cinderella and Prince Charming took up residence in Cinderella’s Castle. With peace throughout the kingdom, Prince Charming had time to practice for jousting tournaments. In the countryside near the castle, he built a training device of carved horses, on which he could practice the art of ring-spearing, a tournament event in which a knight rides his horse full speed, lance in hand, toward a small ring hanging from a tree limb, with the object of spearing the ring. This event was known by various names throughout the lands, but generally came to be called “carrousel.”
The carrousel device drew the attention of the villagers, who wanted to take a turn on this amazing spinning contraption. So Prince Charming had a second carrousel constructed closer to the Castle, where everyone could take a spin on this wondrous invention. Instead of a working knight’s training device, however, this new carrousel is more befitting its regal location in the Castle Courtyard – its rustic training horses replaced with ornately decorated prancing steeds adorned with golden helmets and shields, flower garlands, feathers and other festoons. Prince Charming invites one and all to test their horsemanship skills and to enjoy their own happy ending.