What is second breath in gryphons and how do you know if they have it? i'm confused about this. 

Second breath occurs in gryphons and pumas both, I believe. If it presents as a dominant trait, it gives the animal a 20% chance to gain a second turn in combat - meaning you have a 20% chance to hit the enemy twice before they can hit you.

Other than vermillion - a good way to tell if a gryphon or puma is at least carrying second breath is by looking at their character sheet for an increasted courage score. This means that they are lion-hearted and have the lion-heart trat presenting as dominant. By being lion-hearted, and having the lion-heart trait present as dominant means that no other trait with a higher genetic power is present in the genes of the individual, else that trait would present as dominant instead. However, lion-heart has a higher genetic power than second breath does, so there is a chance that the pet is carrying second breath as a recessive that can be bred and presented as dominant in offspring.

Hope this helps.