Mammoths

Mammoths are much larger relatives of elephants. They are the only species of pachyderm native to Westeros.

Range
Mammoths inhabit the sub-arctic tundras in the far north of the lands Beyond the Wall. They do not venture as far south as the densely wooded Haunted Forest close to the Wall, and as a result they are rarely seen by the Night's Watch.

Biology
Mammoths have a general body plan similar to elephants, though they possess several distinct features, comparable to the difference between Direwolves and regular wolves. Adapted for cold climates, mammoths are covered in long, woolly fur. They are much larger and stronger than normal elephants, and possess disproportionately long, curved tusks, which they can effectively use to combat threats.

Training and riding
Mammoths are used as mounts by the race of Giants, who inhabit the same range of territory far north of the Wall. The size proportion between a giant and a mammoth is similar to that between a normal man and a horse. The giants have not domesticated mammoths, but have been able to tame large numbers of them. Giants use mammoths as war mounts, though they can also be used as beasts of burden.

In the books
In the A Song of Ice and Fire novels, mammoths are ridden as mounts by the giants who live beyond the Wall. The Giants in the novels are very different, inspired by Bigfoot/Sasquatch legends. Covered in shaggy fur themselves, the giants in the books wear no clothing and possess no more advanced technology than wielding logs as clubs (or at most, tying rocks at the end of a club to make a massive, crude but effective maul). The giants' taming of mammoths as mounts is one of their most advanced cultural/technological achievements.

In the books, Jon Snow actually first saw both mammoths and giants at the same time, when he first entered the main wildling army camp under Mance Rayder: a large column of giants was riding into camp at the time, atop their mammoths. The TV series omitted this (probably for budgetary reasons) and only showed a single giant - but hinting at this detail from the books, the giant is seen constructing a tent out of massive mammoth bones.

Regular elephants are fairly common in the warmer parts of Essos, and have been seen in both Slaver's Bay and the southern Free Cities. War elephants are uncommon but not unheard of: the Golden Company possesses its own contingent of war elephants. Volantis, the southernmost Free City, makes extensive use of elephants, as well as using a species of dwarf elephant as both a beast of burden and common means of transport to pull carts, much like horses.

Mammoths are also said to inhabit the icy wastes of the island of Ibben, far north of mainland Essos. It does not appear that mammoths live anywhere on mainland Essos.

Nonetheless, mammoths appear to be the only species of pachyderm native to Westeros itself - making such huge beasts even more exotic and impressive to the men of the Night's Watch when they encounter them.

In real-life
The mammoths in the narrative are essentially similar to the real-life Pleistocene megafauna Mammoths that once existed on real-life Earth. They first evolved about 5 million years ago and spread across both North America and Eurasia. Like real-life Direwolves, however, they died out at the end of the last Ice Age, about 10,000 years ago.

The large Pleistocene mammals went extinct when the last ice age ended: large herbivores like the mammoth died out due to climate change, possibly also combined with hunting by prehistoric humans (humans did hunt mammoths, but how much of a role this played in their overall extinction is debated). Mammoths, like dire wolves, grew to large size as an adaptation for maximizing heat retention, which made them stronger but slower than modern elephants.

For a time, real-life mammoths did inhabit the same areas as real-life dire wolves (who only inhabited North America), just as both direwolves and mammoths inhabit the same area beyond the Wall in Westeros. Real-life dire wolf packs preyed on mammoths, but there have been no explicit descriptions of this occurring in the narrative (mammoths may be very rare and isolated at this point in Westeros's history).