A gene family is a group of genes which share characteristics. Gene families exist because of ancestral duplications; what started out as one gene became many, which moved around in the genome and diverged from each other in the process. Gene families can provide a wealth of interesting information about genetic inheritance and the history of an organism, making them a topic of interest among geneticists, biologists, and other people who are interested in biology and inheritance.
Gene families have strings of similar nucleotides which demonstrate their common inheritance. They may also have a similar process or function; for example, the major histocompatibility complex is a very large and diverse gene family which dictates which antigens will be present on the surface of cells. Together, the members of a gene family can dictate the expression of a trait, whether that trait is a function in the body or an aspect of physical appearance, like a calico coat in a cat.
Members of a gene family can be at different loci, on entirely different chromosomes, and sometimes pop up in a multitude of places. It's also possible to see what is known as a multigene family or superfamily of genes which are not very similar, but which share some key motifs which betray their common ancestry. These gene families are sometimes used to determine when various genetic divergences occurred, as they are typically very old.
Researchers organize genes into families by looking at the DNA they contain and the functions they code for. Organizing into families can provide interesting information; for example, people struggling to understand a particular genetic condition might gain new insight if they realized which gene family was involved. Likewise, being able to lump genes together in a family can help people find out what genes do, as it's possible to tell that a gene is part of a family without specifically knowing what it does.
Gene families also provide information about genetic inheritance. By comparing genetic information from related members of the same species with unrelated members, researchers can see how divergences occurred, and they can sometimes use this data to figure out when they happened. This information can be used to explore the history of a species, by looking for traits which might have arisen in response to changes in the environment or other factors. since changes in the genome tend to occur at a constant rate, people can count backward through the genome to see when changes happened.