There are many different approaches to doing animal photography. It can be a detailed close-up or an action shot of a wild animal; it can also be a casual picture of a pet, or styled in a studio with colourful props. The point is to create an image that connects the viewer to the subject.
Check out this article, called 8 Great Tips for Photographing Animals. It focuses on animals that you might find in a zoo or wildlife reserve, but all of these ideas are easily used when photographing animals in everyday situations. There is also a helpful list called Nature Photographers Code of Conduct. It is a friendly reminder to use common sense and respect when out photographing nature and animals.