Mendelian Genetics
Not much was known about genetics during the time of Mendel, but he believed that "heredity" -- the idea of genetic inheritance -- existed. Rather than assume that traits "blend" together, Mendel realized that probability and a distinct combination of traits would produce certain results for future offspring. Mendel used thousands of species of tall and short plants to prove that a pattern of probability could be figured out based on different traits.
Dominant Alleles
Alleles are basically different forms of the same gene. In gene combinations -- where two alleles exist, one from each parent -- alleles come in two types: dominant and recessive. Eye color is the easiest example of the dominant and recessive mechanism. Brown eyes are considered a dominant trait, possessing a dominant allele. If a person contains two dominant alleles, that is considered "homozygous" for that brown eye gene. Under that condition, the result will always be brown eyes.
Recessive Alleles
On the flip side are recessive alleles. Recessive alleles always take a backseat to their dominant counterpart. Two recessive alleles paired together are considered "homozygous" for that gene, always resulting in the recessive trait. All people with blue eyes are homozygous for the blue eye gene.
Heterozygotes
The real difference comes in "heterozygotes," where a person contains two different alleles -- both a dominant and a recessive one -- for a given trait. In the case of eyes, this means a person that has a dominant brown eye gene as well as a recessive blue eye gene. In this case, the dominant allele will always be expressed, resulting in brown eyes; however, both genes have an equal chance of being passed on to offspring regardless of gene expression. This means that the child -- assuming the other parent has a gene for blue eyes -- has the potential to be homozygous for blue eyes.