This achievement sounds much tougher than it actually is. To unlock the achievement you must raise an animal's happiness to 100%. Happiness is a factor of four elements: Hunger, Cleanliness, Habitat Happiness, and Social Need.
To view the status of an animal:
Edit Habitat
Animals
View Animals
Use LB and RB to sort through your animals.
Hunger is obviously keeping them fed.
Cleanliess is keeping the habitat clean from poop as well as cleaning the animals off themselves.
Habitat Happiness is putting the right type of animal with the right type of habitat as well as putting ammedities in the habitat that fit the animal.
Social Need is meeting the bare minimum for social companions per animal. So if an animal's minimum social need is 3, you need at least 3 animals in the habitat.
Do not worry too much about this achievement as it will almost certainly come naturally as you play through the game.