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.
Rooster by Shawna Cameron

Fall Buckeye Butterflies by Janice McCafferty

Egg Cup Mouse by Sam Clark

Black Cat by sixthandmain

Timid Wandering Penguin by Perfect Art