This is one of the first rules of composition in photography.
Don’t center your subject in your photos. Instead, break your image into thirds both horizontally and vertically (Visualize a grid of three vertical lines and three horizontal ones.) Placing the focal point of your photo along the intersections of those lines will give your photograph more visual interest.

Think about what your focal point is and then remember to visually ‘place it on the grid’. Experiment and see what a difference it will make in your photos.

I find that taking pictures of my own pets is easy, as I am around them all the time and know their moods. But, if you are photographing others pets, you may need to spend some time with them so they become comfortable in your presence.

Taking pictures outside in the natural light is good, if you do the shoot in the early morning or later in the afternoon. Blazing mid-day sun isn’t the best time for a shoot. If you shoot inside, a natural light source (window) is best.

Make sure you get your photos while at eye-level with the animal. This gives your photos a pets-eye view of the world and is much more appealing.

Choose backgrounds that aren’t too busy or cluttered so the focal point is on the pet.

Try to catch them off guard for some fun shots, or use the action/sports mode on your camera to catch them in action.

If the pet is shy and runs from the camera as soon as you bring it out, try to lure them out with a favorite toy, leaving the camera in plain sight. After a while, they may just forget about the camera entirely.

Have fun and experiment with various poses, angles, lighting and backgrounds, you’re sure to get a good shot!