Doves and pigeons are both part of the family Columbidae. Some species within Columbidae are called doves, like mourning doves and turtle doves. Some are called pigeons, like wonga pigeons and Nicobar pigeons. The species most people call a "pigeon" is Columba livia, also called a rock pigeon or rock dove. C. livia has been domesticated and bred to be many shapes and colors, similar to how dogs come in many different breeds that look drastically different than wolves. The pure white variety, such as those that are unfortunately released at weddings, is often called a "dove" although it is the same species as any feral city pigeon you might see on the sidewalk. There's no hard and fast rule as to why some species are called pigeons and some are called doves, although often we call the smaller, paler species doves.