Pigeons, like most life on earth, inherit characteristics form their parents. The traits or characteristics that can be inherited are myriad, from simple eye colour to complex temperament and homing ability. The basic mechanism by which traits are passed down from generation to generation was first studied by Gregor Mendel who studied pea plants in the 1850s. The basic principles he discovered has been explored and refined ever since. This field of study is called [classical] genetics. The next sections explain some of the basics concepts of genetics.
Here is an image to illustrate (a human karyotype):
Karyotype of a female human being (source - Wikimedia)
Genes are responsible for all characteristics of the bird - colour, behavior and body conformation are all controlled by the genes passed down from the parents. Some genes are able to change the characteristics of the bird on their own, while other genes work together to form a specific trait. It is important to remember that some traits are so complex that their expression could be controlled by hundreds or even thousands of genes.
Mutations can happen in a number of ways both natural and artificially induced. Natural mutations occur due to faulty translation of the DNA by a complex mechanism far beyond the scope of classical genetics. Mutations can also be caused by UV rays (e.g. cancerous mutations like melanomas in human beings), harmful chemicals (called carcinogens) or a number of other mechanisms.
Rock Pigeon has more information).
The chromosomal sex determination in birds is known as ZW sex determination. This implies that the female birds have the two allosomes ZW and the males have the allosomes ZZ. (For human beings, this system is reversed, females are XX while males are XY).
Those interested in the mechanisms of sex determination are again urged to have a look at the Wikipedia page .
The next pages give some basic terminology and discuss specific mutations found in domesticated (and feral) pigeons.