These flavors of genes are called Alleles. An allele represents one or more DNA sequences in a gene that produces a particular phenotypic result. Several different mutations might cause a gene to produce a defective protein. All of these mutations might be lumped into a single allele - the gene in a form that produces a defective protein. There isn't necessarily a one to one mapping between mutations to the DNA sequence of a gene and different alleles for that gene.
Alleles are just broad variants of a gene that can produce different results.
Genes come in two copies per individual. The different alleles of a gene may be dominant and recessive with respect to each other. The basic tool for Mendelian genetics is the Punnett Square . It is rare that an aspect of an organism will be controlled by a single gene that has just two alleles. Heredity is seldom this simple.