In genetic terminology, inbreeding is the breeding of two animals who are related to each other. In its opposite, outcrossing, the two parents are totally unrelated. Since all pure breeds of animal trace back to a relatively limited number of foundation pigeons, all pure breeding is by this definition inbreeding, although the term is not generally used to refer to matings where a common ancestor does not occur behind cock and hen in a four or five generation pedigree.
Breeders of purebred livestock have introduced a term, linebreeding, to cover the milder forms of inbreeding. Exactly what the difference is between linebreeding and inbreeding tends to be defined differently for each species and often for each breed within the species. On this definition, inbreeding at its most restrictive applies to what would be considered unquestioned incest in human beings - parent to offspring or a mating between full siblings. Uncle-niece, aunt-nephew, half sibling matings, and first cousin matings are called inbreeding by some people and linebreeding by others.
What does inbreeding or linebreeding (in the genetic sense) do? Basically, it increases the probability that the two copies of any given gene will be identical and derived from the same ancestor. Technically, the pigeon is homozygous for that gene. The heterozygous pigeon has some differences in the two copies of the gene Remember that each bird (or plant, for that matter) has two copies of any given gene (two alleles at each locus, if you want to get technical), one derived from the father and one from the mother. If the father and mother are related, there is a chance that the two genes in the offspring are both identical copies contributed by the common ancestor. This is neither good nor bad in itself.