For every gene in a chromosome, two copies are present. As a matter of ensuring genetic variability among members of a species, different individuals possess different variations of a specific gene. A dominant trait is one that requires only one of the dominant version be present while a recessive trait requires that two copies be present to be seen. Sometimes, the expression of that gene in the offspring physically represents only the dominant copy of the two parents' phenotypes (such as hair color), but the underlying genotype still carries one copy of each parent's genes. For example, black hair color (B) is dominant to red (b). If one parent is BB and the other bb, then each child will carry one of each allele (Bb). The hair color (phenotype) will always be black, but one of each gene is in fact present. If the black haired parent was actually Bb, then some of the children could be Bb (black haired) and others bb (red haired).
Often, the dominant gene represents the heartier variety of that gene, and nature has used this to weed out some carriers of genetic disease from the population. Most if not all disease causing genetic disorders are necessarily recessive. Imagine what would happen if the gene for a deadly disease were dominant (D)- the gene would be expressed in every individual who carried only one copy (both DD and Dd individuals), killing off the race. In contrast, by allowing the healthy form of the gene to be dominant, it is able to "override" the disease-carrying, recessive form, and nature has allowed individuals who carry the recessive gene but also the dominant one to live disease-free and reproduce. If that person were to reproduce with a healthy individual carrying no recessive copies of that gene, then the offspring would have a 50% chance of carrying that gene, but no chance of having the disease. If, on the other hand, the other parent were also a carrier, then the offspring would have a 25% chance of being double dominant (no disease, not a carrier), a 50% chance of being merely a carrier, and a 25% chance of having the disease (a double recessive). That 25% would likely not survive to pass the gene on to others. The carriers also have a smaller chance of passing their disease causing genes to future generations. The end result of all of this is that the disease causing genes stay limited in number, and even further limited in expression, among the population.