There's a Herritage : red star caption under my pet, under energy points (in my stable). What does it mean?

It means that your pet has one grade of Degeneration - so there was inbreeding in its heritage somewhere within the last 3 generations. 

From the FAQ:

"A child gets a Degeneration grade when it is inbred. This means that its parents are relatives within 3 generations. This will result in getting negative genes which parents do not have and the increase of the genetic power of negative genes which they have already. Degeneration grade can't be removed from the pet which already has it but you can decrease or remove it for the future offsprings of this pet by choosing the second parent wisely.

Degeneration always increases or decreases by one grade (except the case of normal breeding and no Degeneration in parents). If the parents aren't close relatives, the Degeneration grade of their child will decrease by 1. If the parents are relatives within 3 generations, the Degeneration grade of their child will increase by 1, the base will be the highest Degeneration grade of the parents."

Thanks! :D