What is a degeneration grade?

It seems I cannot breed some pets due to this.

Thanks for any help :)

You can still breed pets when the degeneration notice goes off, but it's meant to warn you that inbreeding is about to happen (as in a mother and son mating, or a brother/half sister mating, also grandparent to grandchild. Great Grandparents can mate with Great Grandchildren though).

Degeneration results in bad stats being added to your pet, such fragile bones or weak beak. It's generally something you'll want to avoid unless you're breeding deviants and don't care.