A dominant gene, or allele, simply means a gene that alters the phenotype in such a way that it masks the effects of another gene from the same locus. (A locus is the position of the DNA of a gene on a chromosome. The chromosomal locus is a series of numbers detailing the chromosome, arm, band and region of the DNA.)
For example the allele for blue eyes is a recessive trait, and the allele for brown eyes is dominant, meaning that if you have one of each, you will still have brown eyes. The only way to have blue eyes is if you have two copies of the allele for blue eyes, and no copies for brown. Having two copies of one allele and none of the other is called ‘Homozygous’ and having one of each means you are ‘Heterozygous’. So therefore the genotype can be homozygous dominant, heterozygous or homozygous recessive.
There are many reasons why a certain allele is dominant and another is recessive. For example if one codes for a broken protein. The working protein will win out, because the broken protein doesn’t do anything. Other reasons can be simply that one protein is more powerful, as sometimes you need two copies of the gene for enough protein to be made to have an effect. For example the protein CD2AP helps the kidneys filter out harmful enzymes from our blood, and two copies are needed to create enough protein to filter all the chemicals. This means that CD2AP is recessive. MC1R is a dominant gene, as even a tiny bit of it filters all the red pigment from your hair. This means that you must have two copies of the allele for red hair, and no MC1R, to have red hair.
Gregor Mendel was the first man to posit the idea that certain traits would show up more often than others when crossbreeding animals. Using pea plants, he showed that if you cross-bred a yellow pea with a green pea, the offspring was always yellow. He then called the yellow trait dominant and the green trait recessive.
*Note* One allele can be dominant to one, and recessive to another