Donkey
Mule
Bay Plain
Bay White Socks
Bay Paint
Bay Sooty
Bay Leopard
White Plain
White White Socks
White Paint
White Sooty
White Leopard
Black Plain
Black White Socks
Black Paint
Black Sooty
Black Leopard
Grey Plain
Grey White Socks
Grey Paint
Grey Sooty
Grey Leopard
Liver Chestnut Plain
Liver Chestnut White Socks
Liver Chestnut Paint
Liver Chestnut Sooty
Liver Chestnut Leopard
Flaxen Chestnut Plain
Flaxen Chestnut White Socks
Flaxen Chestnut Paint
Flaxen Chestnut Sooty
Flaxen Chestnut Leopard
Buckskin Plain
Buckskin White Socks
Buckskin Paint
Buckskin Sooty
Buckskin Leopard
All of these are different in appearance, but some (especially the white) are extremely hard to tell apart.
I've heard that the blacks are the fastest, the whites have the most health, and the chestnuts jump the highest, but I haven't heard that confirmed. Run speed, health, and jump are the three traits outside of color that can be bred for.