With our current technology, the way we clone animals is to take a somatic cell (not sperm or egg) from the animal we want to clone. The DNA from this somatic cell is combined with an egg cell from the same species of animal, either by injecting the nucleus of the somatic cell into the nucleus-less egg cell, or using electricity to fuse the somatic cell with the egg cell. Then, the completed egg is allowed to grow into an animal embryo in a test tube before being placed into an adult female animal of the same species. When the female gives birth, the resulting animal will have the same DNA as the original animal from which the somatic cell came.

With this method, we have cloned a variety of animals, including sheep, cow, cat, dog, rat, and a few others. These animals were the exact same copies as the animals the cells came from, but we have not been able to make many of these animals because the cloning process is difficult, and the cloned animals require a lot of attention to survive and grow.

To clone an animal, you can take a cell from that animal and transfer its nucleus (the part of the cell containing DNA) to an egg cell from an animal of the same species, and allow that egg to grow into an adult animal. In theory, you could keep cloning the same animal as long as you have cells to borrow a nucleus from!

Different animals do it differently and I'm not sure of the particulars in all case. Most clonal animals do so by budding, where a growth off of an existing animal grows into a new animal. This is how corals and bryozoans do it.

Human technology can clone animals by taking cells from a body and then confusing said cells into thinking that they are embryo cells , and then implanting the embryos into a mother and growing them as children. For any animal with a complex nervous system and even some that don't, the result will not be exactly the same, since the clone will not have any of the learned knowledge of the template.