Your pet makes a great subject to photograph by window light. Find a location where your pet is comfortable that is near a large window. Remember, you want indirect light, no sunlight beaming in for this approach. Control the background by draping it with a plain blanket, sheet or towel that shows off your pet's coat. If you are photographing a dark animal, use a medium or light background. If your pet has a light coat, use a darker fabric.
If your pet likes to sit in this spot frequently, have your camera set up for the right exposure and white balance so you are prepared when your pet decides to sit down. Or you can bring your pet to the location and give them a treat to eat or chew on. Then you can photograph them while they are busy.
In this photograph, the cat is facing toward a large picture window. This provides soft even lighting and a catch light (reflection) in her eyes to help show their color and provide a sparkle of life.