As Vanity said, it would be much easier if you defined what you deemed as 'Animal Magick'. Regardless, I will provide you with some information on relations with animals to the craft.
Animals are often used in symbolism: a variety of animals have a variety of symbolic meanings. For example, if one was to engage within a renewal or persistence ''spell'', one may want to call upon the Hummingbird as it is said to represent such attributes in folklore.
After browsing the internet, I looked at a few sites on animal symbolism, this one by far being the most interesting, detailed and informative:
Now, moving on to other uses. Of course, Animal Totems. Native American animal totems most commonly come to mind when discussing this topic. However, animal signs are found throughout all cultures spanning the globe.
Animal totems aid us in furthering our understanding, aiding us in self-discovery and capture our imagination, giving us incredible avenues of self-expression and awareness. Animals afford us visions of how our lives could be if we lived more simply and lived with purity of thought and emotion. Therefore, incorporating animal totems into our lives affirms our spiritual goals.
By focusing on the attributes of our totems, we internalize these traits and thus begin to externalize the very character we absorb from our totems.
Moving on to another side of the globe, a more Northern tradition, animal fylgja (excuse me if the spelling is incorrect, Norse traditions are something I am only acquainted with).
Originating from the Norse traditions, a fylgja is said to be a supernatural being or creature, usually an animal, which provides company to a person in connection to their fate, fortune, etc.
It is said that fylgjur usually appear during sleep most commonly, however the Norse sagas may have said that they could appear while a person is awake too, and it is rumored that seeing one's fylgja is an omen of one's impending death.
However, when fylgjur appear in the form of women, they are then supposedly guardian spirits for people or clans, which is a rather nice thought. Similarly to the Aztec Nahualli, the animal is usually in an animal form that represents corresponds to the personality and characteristics that it is 'attached' to.
In Norse belief, the fylgja is the embodiment, if you wish, of all of the things the person of whom it 'belongs' to did in the past. It can provide power and knowledge but also quite large challenges and responsibilities that require maturity, to my understanding.
Now, to take from another culture, my chosen culture, Aztec belief- a Nahualli.
Nahualli is a term which denotes two different, yet connected things. In one use, the word nahualli is the term used for a spiritual twin of a person, almost always an animal form. The gods have nahualli as well, in my belief. For example, Tezcatlipoca's (an Aztec god) nahualli form is Tepeyollotl, a jaguar god, and Huitzilopochtli's are the hummingbird and the eagle.
Every person is said to have a nahualli. The nahualli is an animal that you have a spiritual connection to. In this form, a nahualli is much like the concept of a totem animal found in other Native American cultures (as mentioned earlier). Usually, the nahualli has traits that reflect those of the person they belong to.
Unlike the concept of a totem animal, the nahualli denotes an actual individual animal believed to exist, either in the spirit world or in the physical world itself (depending on regional beliefs). The nahualli is the spiritual twin of a person, said to have been born at the same time and share the portion of the soul known as the tonalli.
In the spiritual realms, a person might take the form of their nahualli at times (please do not confuse this with shapeshifting physically, I don't believe in such). In other cases, a nahualli may appear much like the European concept of a familiar animal, showing itself as an individual animal of a person?s nahualli species which the person has a mental and spiritual connection to in this world. While this connection gave strength, it also has the potential to give weakness.