It really depends on what kind of cloning you are referring to. Just the DNA (Polymerase Chain Reaction) or the whole organism.
I'll elaborate on cloning of an animal. An ovum (egg cell) from the organism is extracted and the nucleus with a haploid number of chromosomes is removed. Then a normal cell (like a liver cell or something) with the diploid number of chromosomes, is extracted and its nucleus is transferred to that of the egg cell. This 'zygote' is implanted into the uterus of a third organism and a clone is born.
Cloning organs by themselves is really not possible yet, unless you clone the whole organism and remove its organs. But there are many ethical issues involving cloning, so it probably won't be used on humans for a very very long time.
