You might want to check a genetics text for the difference between a gene and a trait. Wilmer Miller's A Survey of Genetics can be downloaded from his website, ringneckdove.com.
A pigeon cell has 40 pairs of chromosomes. Each chromosome contains a DNA molecule. A pair of chromosomes has identical or very similar DNA molecules. Genes are arranged along the DNA molecule like beads on a string. As the chromosomes are in pairs, the genes are also paired--one gene in one chromosome and the other gene in the other chromosome. A locus is the place in a DNA molecule where a given gene resides. As the genes are paired, a locus has a gene pair.
Ember (symbolized e^em = e with em as a superscript) is a mutant gene that resides at the autosomal e locus. It is recessive to the wild type gene (E^+) at the e locus. Recessive red (e) is another mutant gene that resides at the e locus. Different genes at the same locus are alleles. Therefore, these three genes are alleles. And it is possible to have a gene pair made up of an ember and a recessive red gene (e^em/e).
The e^em/e^em gene pair expresses the ember trait. As far as I know, the e^em/e gene pair also expresses as ember.
Homozygous = The two genes in a gene pair are the same. Examples: E^+/E^+, e^em/e^em, e/e.
Heterozygous = The two genes in a gene pair are not the same. Examples: E^+/e^em (heterozygous ember), E^+/e (heterozygous recessive red), e^em/e (heterozygous ember/recessive red).
Pigeons have 40 pairs of chromosomes. One chromosome pair controls sex determination. They are the sex chromosomes. The other 39 pairs of chromosomes are the autosomes.
In birds, males have two Z sex chromosomes. Females have a Z sex chromosome and a W sex chromosome. Genes with a locus on the Z chromosome are called sex-linked genes. The Z chromosome is large, with many genes. The W chromosome is tiny, with few genes. In the male, the sex-linked genes are paired because there are two Z chromosomes. In the female, the sex-linked genes are not paired because there are no corresponding genes in the W chromosome. The female's single gene at a sex-linked locus is called hemizygous.
The reduced gene (symbolized r) is a mutant gene residing at the sex-linked r (not R) locus. The r locus has two alleles--wild type (R^+) and reduced (r). The male's heterozygous gene pair is R^+/r. As r is recessive to R^+, the bird looks like a wild-type pigeon.
The dilute gene (symbolized d) is a mutant gene residing at the sex-linked d locus. The d locus has four alleles--wild type (D^+), pale (d^P), dilute (d) and extreme dilute (d^ex). Any two of these alleles can form a gene pair. A heterozygous dilute male has a heterozygous gene pair (D^+/d). As d is recessive to D^+, the bird looks like a wild-type pigeon.
Alleles are genes that have the same locus and can form gene pairs. The dilute mutant gene and the reduced mutant gene have different loci and cannot form a single gene pair. Therefore, these two mutant genes are not alleles. A heterozygous dilute, heterozygous reduced male pigeon has two gene pairs (D^+/d R^+/r) and looks like a wild-type pigeon.
Pigeons have something like 17000 loci. All loci (plural of locus) have at least one allele--the wild type allele. This is the allele most commonly found at a given locus in the wild population. Some loci have the wild-type allele and one or more mutant alleles. A mutant gene is any gene that resides at a given locus but is NOT the one most commonly found at that locus in the wild population.
A list of genes includes all alleles at every locus. A list of mutant genes is much shorter. It only includes the mutant alleles at loci with mutant genes.
Opal (o) is a recessive mutant gene found at the autosomal o locus. Stipper or almond (St) is a dominant mutant gene found at the sex-linked St locus. Opal and almond are not alleles.
I'm starting to write a book here.
Wilmer Miller has already done it, and better than I can. For interactions between alleles and between genes at different loci, look at his book and the pedigree problems on his website. Just take it slow and easy. The first time I looked closely at one of his pedigree problems, I thought my brains would melt and trickle out my ears.