Horses, at first, were all wild animals like zebras are today. Although they first evolved in North America, by the time people were spreading through the world, horses had become extinct in North America. All living horses lived in 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 were too small to carry people, and it wasn't until they had been bred bigger that people could ride them.
When the Indo-Europeans began to leave Central Asia and settle in other parts of Asia and Europe, they used their horses to help 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.
Hittite soldiers driving a chariot
Women from Mycenaean Greece driving a chariot, about 1300 BC
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.
Shang Dynasty horses
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.
For more about horses, check out these books from Amazon or from your library:
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.
Or check out the horse article from the Encyclopedia Britannica.