Cloning an animal, or any other organism, refers to making an exact genetic copy of that organism. Two ways to do this are: artificial embryo twinning and somatic cell nuclear transfer.
Artificial embryo twinning is accomplished by manually separating a very early embryo into individual cells, and then allowing each cell to divide and develop on its own. The resulting embryos are placed into a surrogate mother, where they are carried to term and delivered. Again, since all the embryos came from the same zygote, they are genetically identical.
Somatic cell nuclear transfer (SCNT) uses a different approach than artificial embryo twinning, but it produces the same result: an exact clone, or genetic copy, of an individual. In SCNT, the egg cell's single set of chromosomes is removed. It is replaced by the nucleus from a somatic cell, which already contains two complete sets of chromosomes. Therefore, in the resulting embryo, both sets of chromosomes come from the somatic cell.
The lamb Dolly was the first-ever mammal to be cloned and it was done using this method. To make Dolly, researchers isolated a somatic cell from an adult female sheep. Next, they transferred the nucleus from that cell to an egg cell from which the nucleus had been removed. After a couple of chemical tweaks, the egg cell, with its new nucleus, was behaving just like a freshly fertilized zygote. It developed into an embryo, which was implanted into a surrogate mother and carried to term. So Dolly, was an exact genetic replica of the adult female sheep that donated the somatic cell nucleus to the egg.