Damien's pet, a hell hound. The hound was given to him as a present after he (accidentally) threw a very powerful fireball at him at his fifth birthday. The hound helps calm him and helps him channel his powers.
Canine and a solar eclipse. The eclipse doubles as an aureola, suggesting the god-status of the canine. The branches sprouting from the canine's head may have significance as well. I have no idea this means but I love the picture