Any milk from any breed will work nicely for making any cheese. Generally speaking, however, there are certain breeds of cows and goats that are well suited for specific types of cheeses. The choice is based on the size and amount of butterfat globules in the milk they produce.
Cow's milk - Jersey and Guernsey milk has the largest fat globules. It is perfectly suited for the soft and semi-soft cheeses. Ayrshire milk has the smallest fat globules, so it is preferred for the sharp Italian cheeses and long aged Cheddars. Holstein milk is the standard, so it is used across the board.
Goat's milk - Saanen milk is the standard. Nubian milk has larger fat globules, so it is often used for the soft and semi-soft cheeses. Toggenburg milk has smaller globules, so it is traditionally used in sharper, aged cheeses.