A dominant gene is a segment of DNA information that is used to make chromosomes. Dominant genes are a result of dominant alleles in chromosomes. When an allele is dominant it is expressed in the phenotype over a recessive allele. The phenotype is what chromosome is present in the organism. So a person with a dominant allele for brown hair and a recessive allele for red hair (this combination is identified in genetic notation as "Br") will have brown hair.
A child inherits only half of their DNA information from each parent and in this combination generally display the dominant genes they received. If they inherited only recessive genes for some trait, that is the trait that is displayed. This is why siblings can look very different from one another and why a person with two brown-eyed parents can be born with blue eyes (brown eye gene is dominant, blue eye gene is recessive) if they inherited a recessive blue eye gene from each parent.