This is mentioned several times in the books, particularly in "Little Town on the Prairie". It's described as being white and having black spots. It's a bit of a running joke in the book that Pa would often sit on the mantle and talk to the dog, as if it was a real pet.
While it's never explicitly stated what kind of dog it is, it's probably safe to assume it was a common breed of the time, like a terrier or a spaniel.