"The boy's dog is not white."
Translation:El perro del niño no es blanco.
In "The boy's dog is not white.", it implies that it is referring to a specific boy. It's not just any boy, but the boy that owns the dog. So, in conversation a certain boy would be in mind that you would refer to, "el niño", as opposed to "un niño", and so you would use the contraction of "de" (of), and "el" (the), so, " el perro de + el + niño = el perro del niño".
You wouldn't use "del" when referring to a feminine noun. This is because "del" is the contraction of "de"and the masculine article "el", and if you're referring to a feminine noun it's just "de + la", and no contraction. So the phrase would be "El perro de la niña es blanco".