Animals have adaptations that allow them to live in particular climates due to natural selection and gene mutations. For example, polar bears have a thick layer of fat for insulation, and hairs that can transmit sunlight to the dark skin beneath which can maximize the heat they absorb. They have small ears that minimize heat loss to the environment and guard hairs to repel water. Their dense undercoat even covers their feet. In desert climates, water loss is minimized in the urine or feces, or through the skin which is usually waterproof. Heat can radiate out of larger ears as seen in certain rabbits, or by having longer appendages or small bodies. These are but a few examples of adaptations to climate.
An adaptation is something about an animal that makes it possible for it to live in a particular place and in a particular way. It may be a physical adaptation, like the size or shape of the animal's body, or the way in which its body works. Or it may be the way the animal behaves. Each adaptation has been produced by evolution.
As the environment changes, animals that cannot adapt die out, and only the adapted ones survive to produce babies. Because babies are usually more or less like their parents, the whole species soon contains only animals that are adapted to the new environment.
An animal's environment consists of many different things. The climate is important. Whether it is hot, cold, dry, or wet will have an effect on all the creatures that live in a particular place.
Another important part of an animal's environment is what kinds of food plants grow in it. The other animals that live there also have an effect. If there are predators around, the prey animals will have to learn to defend themselves or run fast to escape...etc