Dominant alleles are those that are expressed in the phenotype of an individual, even if the individual also has one copy of the recessive allele. For example, if a gene controls eye color, and the dominant allele is for brown eyes, then an individual with one copy of the brown eye allele and one copy of the blue eye allele will have brown eyes.
Recessive alleles are those that are only expressed in the phenotype of an individual if the individual has two copies of the allele. For example, if the blue eye allele is recessive, then an individual would need to have two copies of the blue eye allele in order to have blue eyes.