Pets tend to get a little lost in photos unless you make an effort to get up close to them, which sometimes is difficult, especially if you have a pet that won’t be still. But, it is worth the effort! A detailed, focused photo shows the specific characteristics of your pet. If getting close isn’t an option, use a zoom lens as this will isolate your pet as the star and the background won’t matter, since it’ll blur regardless.
Get down on your pets’ level where you look at them eye to eye. Photos taken from above the pet leave the subject too far away and at an odd angle for portraiture. Even a series of close up facial feature shots (eyes, nose, ears) in addition to whole face ones are helpful to show your unique pet.
This is key. I don’t recommend using a flash for a few reasons. Not only can it scare the animal away from looking at you, but it creates the dreaded “red-eye”. Natural light is always best if possible. However, if your pet is very dark in color, a flash can help distinguish fur details. Alternately, an all-white dog gets washed out with a flash/very direct sunlight.
If your camera has a shutter and or sports mode, give it a try! This allows more frames shot in succession, which might capture “the” image, especially if the pet moves.