Gene dominance is complicated and can vary by gene. For the example that you give, eye color, the allele for the brown and blue phenotypes make more (brown) or less (blue) melanin pigment. The blue allele makes little to no melanin creating colorless irises that refract light, like the sky, creating shades of blue. Brown alleles produce more pigment and, therefore, more color.
Why are some genes dominant over others? What is the mechanism behind it?
2 Answers By Expert Tutors

Noah L. answered 05/23/19
High School and College Tutor Specializing in Sciences
There's a phenomena called "teaching to kids" wherein you have to over-simplify material in order to make it digestible for learners before adding nuance. An example of this is how when you first learn chemistry, you start with the periodic table, then move to the Bohr model (even though it is inaccurate), then electron clouds, then quantum states.
The kids-version of molecular genetics has all genes acting in a monogenic, dominant-recessive, non-epistatic, non-pleiotropic way. But that's not really true for 99.9999% of genes. Most genes are polygenic, most are pleiotropic. And dominance is more nuanced because there's co-dominance, incomplete dominance, and epistatic effects.
So, then it essentially comes down to proteins. Your brown eyes are brown and not blue because all the genes that are collectively responsible for encoding RNA that will be transcribed into proteins that create pigment... well, quantitatively there is a threshold that is reached to make your eyes "brown", and not "blue". You have more of those genes that are brown than are blue. Or you have genes that are suppressing the expression of blue pigment creation.
And I say quantitatively because, not all brown eyes are the same. All the genes that collectively code for the protein pigments are different for each person. (Eye color is quantitative like height).
Hope that helps.
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