There are three primary types of cloning, (1) recombinant DNA technology or DNA cloning, (2) reproductive cloning, and (3) therapeutic cloning.
Therapeutic cloning is the one scientists hope will be successful for organ cloning. This would be done by extracting DNA from the person receiving the transplant that DNA is inserted into an enucleated egg. After the egg (now with the donors DNA) begins to divide, the embryonic stem cells are harvested. These are the cells that can be developed in to any type of cell. Those cells can can then be grown into the complete organ or tissue for the donor and will be a full genetic match (in theory). This organ cloning would eliminate the need for anti-rejection drugs than can cause some many problems with donor recipients.
There are still several obstacles to overcome before organ cloning is a reality. The technology for creating human embryos, harvesting stem cells, and producing organs from stem cells is not efficient.
Another possibility for organ cloning is to create genetically modified pigs from wich organs suitable for human transplants could be harvested. This kind of transplantation is called xenotransplanation since it is from animal to human. Although Primates are a closer match to humans, they are more difficult to clone and their reproduction rate is lower. Of the species that have been cloned to date, pigs create the organs most similar to humans.
The technical and moral debate over organ cloning will continue for years to come. It is almost certain that organ cloning will eventually become a reality in some countries.