Genetic traits are passed from parents to their offspring in the form of genes. We can think of genes as a data lookup table read by the software of life to determine what traits to develop.
Each gene comes in pairs of alleles. It can have one of two possible chemical structures, which we geeks can think of as a bit of digital data. Its value can be either 0 or 1.
Now, the reason for the genes coming in pairs is to allow us to inherit a genetic bit of data for each gene from both the father and the mother. Naturally, the father and the mother have genes which come in pairs, just like ours do. For each gene, one allele comes from the father, one from the mother.
Furthermore, which of the two alleles is inherited for each gene is "decided" randomly. That is why siblings' genetic code is similar but not identical.
If we continue the 0/1 model of genes as a lookup table, then each gene can have the binary value of 00, 01, 10, or 11. How the "software" interprets these values varies from gene to gene (though it is always the same for the same gene).
Most commonly, genes are interpreted according to the recessive/dominant formula (the other common formula uses the "incomplete dominance" model). Mathematically, this uses Boolean or in which the recessive gene equals to Boolean 0, the dominant gene corresponds to Boolean 1.
That results in the following truth table:
00 = 0
01 = 1
10 = 1
11 = 1
In other words, whenever the dominant gene is present, it "wins". The recessive trait only manifests if inherited from both parents.
That makes the recessive trait rarer than the dominant trait. Assuming both 0 and 1 are equally distributed, the recessive trait will manifest in only 25% of the population, even if it is present in the genes of 75% of the population.
Interestingly, there is a philosophical point hidden there somewhere suggesting that the gentle, the weak, the soft, the meek, and such, is better fit to survive than the strong, the powerful, the arrogant, the pushy, and the like.
Consider a madman keen on destroying a dominant trait, for example, some Hitler killing off all people with brown eyes (dominant trait). That would be the end of brown eyes. If, on the other hand, he killed off all blue-eyed people, the next generation of children would promptly restore blue eyes.
NOTE: As the mathematician said in Jurassic Park, nature finds a way. There is a genetic phenomenon called mutation that would eventually restore brown eyes to existence.