What do the red and green stars mean in the heritage?

The Green star means Genetic Power (you get them by breeding pets at level 50, or if the pet has GP), and the Red stars are Degeneration 

Anyways, you can read up about it here:

http://beastkeeper.com/faq -- The Faq,

http://beastkeeper.com/threads/21 -- The Player Guides section (I'm too lazy to look for the links right now, like how you're too lazy to do the same, thus asking here.