What do the different gene and mutation trait colors mean? What does the purple color traits mean as far as breeding goes?

Green= carried trait
Blue= visable trait
Purple= visable with recessive hidden trait

The above information is correct, but to add to it -

Purple, as mentioned above, signifies a visible trait with a hidden trait behind it. This means there is a more recessive trait that you can't see on the pet, but it can be passed onto offspring.

For example, if a pet has AB in the G1 slot, its offspring can inherit either A or B. If the pet had AA, on the other hand, the offspring could only ever inherit A.