The best photos give you a glimpse into your pet's life, to see things from her perspective. You won't get that by standing over her and shooting down at the floor. Crouch or lay down, and put your camera at her eye-level.
Get as close as you can to your pet without bothering him. Show him, not the rest of the room. For the photo here, I was no more than three feet away.
A photo of your pet sitting down is a tall triangle, and a head-and-shoulders photo is a tall rectangle. Orienting the camera vertically will help you zoom in and eliminate background clutter.
You know your pet well. You know when she will want to take a nap, how she will react to your child coming home from school, or what she will do when she sees her favorite toy. Take advantage of that knowledge and be ready with the camera when those things happen.
Photographers obsess about light because it can be the difference between a ho-hum photo and a great photo. Low-angle light at sunset/sunrise is warm and golden, and will create highlights that glint off your pet's fur, and soft shadows that provide depth and texture. Mid-day light is blue and flat, and not usually flattering. It goes without saying that using natural light is almost always better than using the flash.
Changing your camera’s automatic settings can give you a far better photograph in certain situations. For the photo above, my camera initially chose an exposure that would have shown all the background furniture, which would have spoiled this shot. Instead, I forced the background to be underexposed, which let Simon appear spotlighted against a plain black background. There are only a few things to learn: shutter speed (how much time you give the camera to capture the image), aperture (how much light you let into the camera), and ISO (sensitivity to light). Spend some time with a good photography book or website, and practice what you’ve read. You'll love what you can do!