Horses, at first, were all wild animals like North America about 3.5 million years ago, horses became extinct there. Probably when buffalo arrived in North America from Central Asia, about 10,000 BC, they ate the grass the horses needed. Then people probably hunted North American horses to extinction around 5600 BC. All living horses are descended from the horses that migrated to Central Asia, where they ate the long grass that grew there, and also the native apples and carrots (that’s why horses love apples and carrots even today!).
When the first people arrived in Central Asia, about 100,000 BC, they hunted horses for their meat and especially for their skins, to make into leather hides for clothes and for tents and tools. But around 4000 BC, people in Central Asia began to tame horses, to domesticate them, to eat them and to use them to carry things. It was probably the Indo-Europeans, living around the Caspian Sea in Central Asia, who first tamed horses for their own use. The first horses may have been too small to carry people; they pulled wagons instead (but it’s also possible that some people were riding horses).
Soon the idea of using horses and wagons to carry people and stuff began to spread out of Central Asia. By about 2500 BC, Sumerian people in West Asia were using horses and wagons. Horses could carry trade goods from one city to another, and they could pull wagons full of people or hay or wheat or pots from one place to another too.
When the Indo-Europeans began to leave Central Asia and settle in other parts of Asia and Europe, their horses helped them win their battles. The first appearance of the horse in Greece comes with the arrival of the Indo-Europeans around 2100 BC. The first appearance of horses at Troy is around 1900 BC, also probably with the arrival of the Indo-Europeans. And the first arrival of the horse and chariot in Egypt comes with the invasion of the Hyksos, or Amorites, around 1700 BC, when the Amorites had been learning things from the Indo-European Hittites.
By about 1200 BC, in the late Shang Dynasty, people in China were also using horses and chariots. This grave from China (from about 1200 BC) contained two horses, a chariot, and their charioteer, who were all sacrificed for the grave of a rich and powerful man.
Having tame horses made a big difference to people’s lives. First off, horses were a tremendous military weapon. You could use chariots to get into battle and use them to squash your enemies, and you could ride them in order to get from one city to another much more quickly than the other army could. You could send quick messengers. And you could carry tents and food on their backs.
Eyewitness: Horse, by Juliet Clutton-Brock (2000). For kids, with lots of pictures.
The Horse in the Ancient World, by Ann Hyland (2003). Mostly Greece and Rome.
Imperial China: The Art of the Horse in Chinese History, by Bill Cooke (2000). This is the catalogue of an art exhibit, so it has lovely pictures of everything to do with horses in imperial China. It has a lot of information about the history of horses in China, too.