Light coming from a single direction creates interest. In this way one side of the face is lighter, while the other is darker. Natural light is the preferable to interior light. Catching your pet by a window, like a cat sunbathing on the back of the couch, can be great to catch light coming in from one direction. Window light is also nice because it is less bright than FULL sun where the contrast may be too great. A covered porch or an open door also work really well. Please do not use a flash.
The photo of Roxy, the lab above, was an Iphone pic taken in an open garage. It worked perfectly!
Try to capture a good shape or silhouette, like your pet looking directly at you or off to the side.
It can take a BUNCH of tries to get one good shot! Don't give up. Getting home with your phone camera on and ready can be an easy way to catch a shot of an excited pet coming to greet you. Try pulling the best frame from a video or turn up your shutter speed to take multiple frames per second like a paparazzi.
If your dog is bouncing around, try putting him on the leash and tying him up. Then walk back a bit and shoot from 5-10 feet away. Remember to try different eye levels- some down on a knee some standing taller and take more than you think you need.
On an overcast day, this picture taken of Fester outside on the patio creates subtle light and dark sides which can make a great reference.
If your pet is on the move, make sure your camera can catch him in focus with good resolution. Try a sports mode with 3 frames/second or shoot video and pull the frames out after. Also consider enlisting the help of a friend. Then just cross your fingers they run into some good lighting!