Horses. There are a lot of them in Rohan. In fact, "Rohirrim" (the term for people living in Rohan) means "horse-lords." The open fields and vast plains of Rohan make it an ideal landscape for traveling or fighting on horseback. Even some of the Rohirrim names are horse-related. It's something which distinguishes the Rohirrim from their fellow men of Gondor—what are the lords of Gondor known for? A white tree? Pshh.
But when is a horse more than a horse? Answer: all the freaking time.
Just think about western culture in general. Even outside of Tolkien's mythology, horses aren't just small elephants or big dogs, they're creatures set apart from the rest. Ever heard of the phrase "wild stallion" used to describe a particularly, or dangerously, free spirited person? Horses need open spaces to roam and run and graze in. They're an embodiment of power and freedom… and by riding horses, the people of Rohan capture this spirit themselves.
Of course, there are special horses too. Aragorn's Brego, whom he releases from the stable and who later finds him on the riverbank, seems to have a special bond with him despite his wild nature. And then there's Gandalf's Shadowfax; "the lord of all horses," Gandalf calls him, "has been my friend through many dangers."
Shadowfax was a descendant of the Mearas, an intelligent and powerful breed. Like Brego, he also has a connection to the wizard that transcends tameness. He's loyal—come on, he even comes back for Gandalf after Gandalf battles the Balrog and becomes the new-and-improved White Wizard.