Special genes refer to the patterns, colors and otherwise odd genes that are not a result of being a particular species. There are genes that are specific to some species that may be special for others--such as bobtail!
The genetics work in simple ways. Many genes work via simple dominant/recessive relationships. This includes:
Others are "dosage dependent" IE they are incompletely dominant
Some require multiple genes.
Some are an unstable allele. In the case of albinism, the gene can randomly mutate (about 1/100 breedings).
Some have other, unintended consequences, such as starburst.