The majority have to be unlocked through the different challenges. In the game, you can see the challenges by hitting the START button on the controller and then X when the Challenge List is highlighted, or flashing.
Others can be bought as DLC, and they include: two other Pomeranians, Panda, Saber Cat, Giraffe, Fat Cat (larger, stronger version of the cat), Silky Terrier, Crocodile. A few more can be unlocked by completing Story mode, or getting them through one of the various ones mentioned that can be bought. Each time you unlock the ones via challenge completion, it uses your survival points. Some of the animals have various colorations or other genders (such as the cats with color variants and Blackbucks with gender choices). Some stop after a certain point, meaning it doesn't move on to another animal such as the Chimpanzee, one of the dinosaurs (can't remember its name) and the Polar Bear.