An unfertilized donor egg cell is taken from another animal and has its nucleus replaced with one from a cell from the animal to be cloned. The reprogrammed egg cell is then stimulated to start dividing much like a normal fertilized egg.
The original cells were frozen in 1988 and since then the population of black-footed ferrets has dwindled to just a few individuals all living in captivity. The article mentions that all black-footed ferrets alive today are essentially half-siblings, except for this clone.