Genes come in pairs. If a pair contains two different genes, typically one of them is dominant and the other is recessive. This means that the characteristic or trait supplied by the dominant gene will be manifested. The only way the characteristic or trait supplied by a recessive gene can be manifested is for both of the genes to be the recessive one.
As has been noted, in humans the gene for brown eyes is dominant whereas the gene for blue eyes is recessive. This means that only people with two blue-eye genes will have blue eyes, and the rest will have brown, regardless of whether they have one or two brown eye genes. (I'm simplifying; obviously there are other genes for green eyes, etc.) This means that most people will have brown eyes, even if quite a few of us might have one blue eye gene.
The same is true of hair. Brown hair is dominant, blond is recessive. So people with only one blond hair gene will have brown hair, even if there are a whole bunch of them. Again, blond hair is unusual in most human populations, regardless of how many members have one copy of the gene.
If the dominant gene is a useful one, the prevalence of the trait it programs will be beneficial to the species. If it is not useful, it might endanger the species. At the very least, all individuals with that gene--either one or two--are likely to die off, and the species will be left with only the individuals with a paired-up recessive gene.
This is an unstable equilibrium, of course. Over the generations, a mutation may occur, resulting in the emergence of a new gene, which happens to be dominant. As the organisms breed with each other, this new gene will spread throughout the population. If it happens to manifest a dangerous trait (for example, bright red fur on an arctic rodent so all the predators can spot it easily), it could result in the species dying off.