Try to position your subject so that the light source shines directly into their eyes. If the source is too strong for this, try to get the subject to look askew while the light source is still beaming toward its face. Solid color backgrounds are often the best, though it is critical that the background be a color and texture that the subject will not blend into. The idea is to showcase the subject, not camouflage it. A green parrot in front of a green plant, for instance, probably won't cut it.
Some pets are just harder to photograph than others.
Dark-colored pets can be a challenge. All-white animals can also be tough. Medium to light shades are often the easiest to capture nicely on film, thus the volume of great shots of pets such as Golden Retrievers. Small animals are usually harder to photograph than larger pets. In getting close to shoot, say, a mouse, focus becomes critical and tougher to control.