The Book of Mormon is very clear that there were particular animals among the Nephites and Lamanites. Thanks to paleontologists and other researchers, we now know which animals actually did live in the Americas during the Book of Mormon period of history.
Here are good summary verses from the Book of Mormon:
1 Nephi 18:25
And it came to pass that we did find upon the land of promise, as we journeyed in the wilderness, that there were beasts in the forests of every kind, both the cow and the ox, and the ass and the horse, and the goat and the wild goat, and all manner of wild animals, which were for the use of men. And we did find all manner of ore, both of gold, and of silver, and of copper.
Ether 9:18–19
And also all manner of cattle, of oxen, and cows, and of sheep, and of swine, and of goats, and also many other kinds of animals which were useful for the food of man.
And they also had horses, and asses, and there were elephants and cureloms and cumoms; all of which were useful unto man, and more especially the elephants and cureloms and cumoms.
To get right to the point — cows, oxen, asses, horses, domesticated goats (as opposed to wild goats as mentioned), swine, and elephants were not present in any part of the americas during any time even close to around the the Book of Mormon history (2500 BC to 400 AD, roughly). And this isn’t just a guess, it is well understood which animals were are were not present during that time.
Apologetics (which are defenders of the faith, not apologizers) respond that the cows, oxen, asses, horses, goats, swine, and elephants were not really cows, oxen, asses, horses, goats, swine, or 
elephants. Instead, Joseph Smith, Jr used those more familiar words to refer to whatever other animal was actually around in the Book of Mormon time, such as bison, deer, tapirs (see picture), or llamas.
I just am unable to accept this answer. First, there is no evidence of domestication of bison, deer, or tapirs — there is only evidence of the llama being domesticated in pre-Columbian America. The Book of Mormon verses above and others clearly demonstrate that the animals were completely domesticated.
Secondly, and I think maybe more importantly, based on how we understand the translation process, Joseph Smith would basically read the words that would appear in front of him. He started with the Urim and Thummim, then he switched and translated most of the Book of Mormon by reading the words off the peep stone he found in his youth, while putting his head and peep stone inside a hat. See Martin Harris’s and David Whitmer’s descriptions of it here. If he was being shown the words to say, would he be shown the word “horse” for something that wasn’t horse? Why would other times unknown names appear, like the “cureloms” and “cumoms”, or “ziff” (an unknown metal) and “senine” (a measurement)? If it wasn’t referring to a known animal, shouldn’t he have read some other word like “cureloms” when he read “horse” or “cow” or “swine”?
Let’s dig into a few of these some more.
Horse and Chariot
From wikipedia:
Horses are mentioned fourteen times in the Book of Mormon, and are portrayed as an integral part of the cultures described. There is no evidence that horses existed on the American continent during the 2500–3000 year history of the Book of Mormon (2500 BC–400 AD). Horses evolved in North America, but are believed to have become extinct on the American continent at the end of the Pleistocene. Horses did not reappear in the Americas until the Spaniards brought them from Europe.They were brought to the Caribbean by Christopher Columbus in 1493, and to the American continent by Cortés in 1519.
Here are the verses in the Book of Mormon:
1 Nephi 18:25
And it came to pass that we did find upon the land of promise, as we journeyed in the wilderness, that there were beasts in the forests of every kind, both the cow and the ox, and the ass and the horse, and the goat and the wild goat, and all manner of wild animals, which were for the use of men. And we did find all manner of ore, both of gold, and of silver, and of copper.
Enos 1:21
And it came to pass that the people of Nephi did till the land, and raise all manner of grain, and of fruit, and flocks of herds, and flocks of all manner of cattle of every kind, and goats, and wild goats, and also many horses.
Alma 18:9
And they said unto him: Behold, he is feeding thy horses. Now the king had commanded his servants, previous to the time of the watering of their flocks, that they should prepare his horses and chariots, and conduct him forth to the land of Nephi; for there had been a great feast appointed at the land of Nephi, by the father of Lamoni, who was king over all the land.
Alma 18:10
Now when king Lamoni heard that Ammon was preparing his horses and his chariots he was more astonished, because of the faithfulness of Ammon, saying: Surely there has not been any servant among all my servants that has been so faithful as this man; for even he doth remember all my commandments to execute them.
Alma 18:12
And it came to pass that when Ammon had made ready the horses and the chariots for the king and his servants, he went in unto the king, and he saw that the countenance of the king was changed; therefore he was about to return out of his presence.
Alma 20:6
Now when Lamoni had heard this he caused that his servants should make ready his horses and his chariots.
3 Nephi 3:22
And it came to pass in the seventeenth year, in the latter end of the year, the proclamation of Lachoneus had gone forth throughout all the face of the land, and they had taken their horses, and their chariots, and their cattle, and all their flocks, and their herds, and their grain, and all their substance, and did march forth by thousands and by tens of thousands, until they had all gone forth to the place which had been appointed that they should gather themselves together, to defend themselves against their enemies.
3 Nephi 4:4
Therefore, there was no chance for the robbers to plunder and to obtain food, save it were to come up in open battle against the Nephites; and the Nephites being in one body, and having so great a number, and having reserved for themselves provisions, and horses and cattle, and flocks of every kind, that they might subsist for the space of seven years, in the which time they did hope to destroy the robbers from off the face of the land; and thus the eighteenth year did pass away.
3 Nephi 6:1
And now it came to pass that the people of the Nephites did all return to their own lands in the twenty and sixth year, every man, with his family, his flocks and his herds, his horses and his cattle, and all things whatsoever did belong unto them.
3 Nephi 21:14
Yea, wo be unto the Gentiles except they repent; for it shall come to pass in that day, saith the Father, that I will cut off thy horses out of the midst of thee, and I will destroy thy chariots;
Ether 9:19
And they also had horses, and asses, and there were elephants and cureloms and cumoms; all of which were useful unto man, and more especially the elephants and cureloms and cumoms.
As children, we were all taught in American History classes about the profound impact that horses had on the Indians once they were introduced to the New World by the Europeans. I have a hard time believing that all the history books, scientists, Indian records, etc. are all wrong about something that would have been so important to the Native Americans.
The chariots are a huge problem as well. As stated in wikipedia:
The Book of Mormon mentions the use of chariots as a mode of transportation five times. There is no archaeological evidence to support the use of wheeled vehicles in Mesoamerica.
Many parts of ancient Mesoamerica were not suitable for wheeled transport. Clark Wissler, the Curator of Ethnography at the American Museum of Natural History in New York City, noted: “we see that the prevailing mode of land transport in the New World was by human carrier. The wheel was unknown in pre-Columbian times.”A comparison of the South American Inca civilization to Mesoamerican civilizations shows the same lack of wheeled vehicles. Although the Incas used a vast network of paved roads, these roads are so rough, steep and narrow that they appear to be unsuitable for wheeled use. Bridges that the Inca people built, and even continue to use and maintain today in some remote areas, are straw-rope bridges so narrow (about 2–3 feet wide) that no wheeled vehicle can fit (see image and technology at Inca rope bridges). Inca roads were used mainly by chaski message runners and llama caravans.
The picture shown here is also from wikipedia. It is a typical Inca road, as described in the prior paragraph. The use of the chariot behind a horse as a mode of transportation between cities is clear in the Book of Mormon verses. This is just not possible. There was no horse, there was no wheel, and there was no suitable road.
Cattle and Cows
Here are the relevant verses:
1 Nephi 18:25
And it came to pass that we did find upon the land of promise, as we journeyed in the wilderness, that there were beasts in the forests of every kind, both the cow and the ox, and the ass and the horse, and the goat and the wild goat, and all manner of wild animals, which were for the use of men. And we did find all manner of ore, both of gold, and of silver, and of copper.
Enos 1:21
And it came to pass that the people of Nephi did till the land, and raise all manner of grain, and of fruit, and flocks of herds, and flocks of all manner of cattle of every kind, and goats, and wild goats, and also many horses.
Mosiah 13:18
But the seventh day, the sabbath of the Lord thy God, thou shalt not do any work, thou, nor thy son, nor thy daughter, thy man-servant, nor thy maid-servant, nor thy cattle, nor thy stranger that is within thy gates;
3 Nephi 3:22
And it came to pass in the seventeenth year, in the latter end of the year, the proclamation of Lachoneus had gone forth throughout all the face of the land, and they had taken their horses, and their chariots, and their cattle, and all their flocks, and their herds, and their grain, and all their substance, and did march forth by thousands and by tens of thousands, until they had all gone forth to the place which had been appointed that they should gather themselves together, to defend themselves against their enemies.
3 Nephi 4:4
Therefore, there was no chance for the robbers to plunder and to obtain food, save it were to come up in open battle against the Nephites; and the Nephites being in one body, and having so great a number, and having reserved for themselves provisions, and horses and cattle, and flocks of every kind, that they might subsist for the space of seven years, in the which time they did hope to destroy the robbers from off the face of the land; and thus the eighteenth year did pass away.
3 Nephi 6:1
And now it came to pass that the people of the Nephites did all return to their own lands in the twenty and sixth year, every man, with his family, his flocks and his herds, his horses and his cattle, and all things whatsoever did belong unto them.
Ether 9:18
And also all manner of cattle, of oxen, and cows, and of sheep, and of swine, and of goats, and also many other kinds of animals which were useful for the food of man.
Here is the wikipedia information:
There are six references to cattle made in the Book of Mormon, including verbiage suggesting they were domesticated. There has been no evidence recovered that Old World cattle (members of the genus Bos) inhabited the New World prior to European contact in the sixteenth century AD.
Apologists argue that the term “cattle” may be more generic that suggesting members of the genus Bos, and may have referred to bison, mountain goats, llamas, or other American species. According to the Book of Mormon, varieties of “cattle” (including goats and sheep) could be found in ancient America. Without these the Nephites could not have kept the Law of Moses, as directed.
Latter Day Saint apologists note that the word “cattle” may refer to the ancestor of the American bison, Bison antiquus (of the subfamily Bovinae). Bison antiquus, sometimes called the ancient bison, was the most common large herbivore of the North American continent for over ten thousand years, and is a direct ancestor of the living American bison.
However, no species of bison is known to have been domesticated as the “cattle” in the Book of Mormon are suggested to have been. Furthermore, it is widely accepted that the only large mammal to be domesticated in the Americas was the llama; no species of goats, deer, sheep, or other “cattle” were domesticated before the arrival of the Europeans to the continent. Apologists counter that the wording in the Book of Mormon does not require the “cattle” to have been domesticated in the strictest sense.
There isn’t much else to say here. There had to be domesticated cows, sheep, and goats as claimed in the Book of Mormon in order to keep the Law of Moses, as claimed in the text. There were none in pre-Columbian America.
Elephants and Swine
From wikipedia:
Elephants are mentioned twice in a single verse in the Book of Ether. Mastodons and mammoths lived in the New World during the Pleistocene; however, as with the prehistoric horse, the fossil record indicates that they became extinct along with most of the megafauna towards the end of the last ice age. The source of this extinction is speculated to be the result of human predation, a significant climate change, or a combination of both factors. It is known that a small population of mammoths survived on St. Paul Island, Alaska up until 8,000 B.P., but even this date is thousands of years before the Jaredite record in the Book of Mormon begins.
Swine are referred to twice in the Book of Mormon, and the narrative of the Book of Mormon suggests that the swine were domesticated by the Jaredites. There have not been any remains, references, artwork, tools, or any other evidence suggesting that swine were ever present in the pre-entrada New World.
Similar to the prior sections, these animals were not here in Book of Mormon times, and furthermore there is no evidence of domestication of any even vaguely similar animals.