I will try to start with very basics. My apologies if I am too simple.
Every creature with two parents has two sets of chromosomes: one inherited from each parent. Except for the sex chromosomes, the chromosomes are matched pairs, each containing locations for specific genes. These non-sex chromosomes are referred to as being autosomal.
Within each location on a chromosome, a set of specific variations can occur. Sometimes there are only two alternatics, sometimes many. These variations are referred to as alleles. For the blue gene (since that seems to be the most commonly explained) there are two alleles: blue and not-blue. Blue is abbreviated Bl, using a capital letter because it is dominant over the alternative allele, not-blue, which is abbreviated bl+. The + indicated that this allele is wild-type. Because chromosomes are paired, each bird carries two copies of a blue allele. They can be two copies of Bl, two copies of bl+ or one of each. When both alleles are the same, it is referred to as being homozygous; when they are different it is heterozygous.
So, each location holds a different gene, of which there can be several variations.
Kind of like a row of houses, each with a different family. Only the Adams live at the Adams' house, only the Browns live at the Brown's house, only the Carlisles live at the Carlisle's house. But, at the Adams there can be Mr Adams, or Mrs Adams, or Baby Alex Adams; at the Browns you might find Aunt Beatrice Brown, Uncle Bob Brown or Cousin Betty Brown. etc.
Transfering this analogy to chicken genes, Dominant White would be one house, and living there are Smokey (I^S) Dominant White (I), Dun )I^D), and not-White (i+). Another house is home to the Recessive Whites: Colour (C+), not-colour (C), red-eye (C^re) and albino (c^a). And yet a third house is home to the Extended Blacks: Extended (E), Birchen (E^R), Wheaten (e^Wh), not-extended (e+) and Brown (e^b). Now the rules of this little universe are somewhat complex, but everyone has an identical twin who lives at their home. Also, in most cases, exactly two of the family is always home. In a few cases for a few homes, only one person can be at home.
As in all families, each gene-family has their own traditions and rules, and each family member has their own personality.