JOEL - It has been known for a long time that the Equus species of the horse existed at one time in the western hemisphere, but it has been assumed that the species became extinct thousands of years before the time frame of Book of Mormon events.
The Smithsonian in their "Statement Regarding the Book of Mormon" prepared by the Department of Anthropology, National Museum of Natural History, Smithsonian Institution, Washington, DC SIL-76, Summer 1982 reports: "American Indians had no ... horses, donkeys, camels before 1492."
There are basically two possibilities that explain the mentioning of horses in the Book of Mormon. The first, is that the horse did not become extinct as long ago as was previously thought.
Actual horse bones have been found in a number of archeological sites on the Yucatan Peninsula. These remains, as well as cave art depicting various horse shaped figures with people riding on them, have been discovered along with other artifacts that date these findings long after the horse supposedly became extinct and long before the Spaniards brought their horses to the American continent. Critics often state that the lack of hard evidence of horse remains, proves that there were no horses at the time during the Book of Mormon history. However, you can not prove a negative. One can not say that something does not exist simply because it has not been found yet. So far only a very small fraction of earth has been uncovered in the western hemisphere during archeological research. And in most of that research, scientists have not been specifically looking for horse remains. There are still many more places to look.
The second possibility is that, what was called a horse in the Book of Mormon, was not the same animal we call a horse today. When the Nephites came to this continent, they found animals here that they had never seen before. They may have given them names that best described the animals based on what they were familiar with. What they called a horse may have actually been nothing more than what we would call a deer or a tapir. The Mesoamerican variety of tapir(tapiris bairdii) can grow to be nearly six and a half feet in length and can weigh more than six hundred pounds. Many zoologists and anthropologists have compared the tapir's features to those of a horse or a donkey. Zoologist Hans Krieg notes, "it reminded me of an animal similar to a horse or a donkey. The movements as well as the shape of the animal, especially the high neck with the small brush mane, even the expression on the face, are much more like a horse's than a pig's [to which some have compared the smaller species]. When watching a tapir on the alert . . . as he picks himself up when recognizing danger, taking off in a gallop, almost nothing remains of the similarity to a pig."
No mention is made in the Book of Mormon of people riding on their "horses" and at one point the animals are mentioned along with other groups of cattle and flocks as though they were used as a source of food rather than as transportation(3 Nephi 4:4), although they were apparently used for pulling chariots(Alma 20:6). The confusion of what this animal should have been called may have carried through during the translation of the Book of Mormon as well. The Hebrew word for horse , "sus", has a root meaning of "to leap" and could therefore refer to any other animal that fits this description.

For more extensive information on this subject please see the following sites:

Questions about Missing Plants and Animals in the Book of Mormon - by Jeff Lindsay

Horses and The Book of Mormon - Compiled By Glen W. Chapman

Book of Mormon Anachronisms - by Michael R. Ash

And this book:
"An Ancient American Setting for the Book of Mormon" by John L. Sorenson." Deseret Book Comp., SLC, UT (1985).