As with any portraiture, cluttered backgrounds are out, lines through heads are out, distracting features unless it helps tell a story.
Natural settings work well with animals, ie the cat out on a limb of a tree close up, stalking in the long grass front on, playing with some small object.
Lighting is a big part, flash shadows are out, fill flash should be soft and discrete, natural light if you are outdoors like landscape photography that special early morning light.
If you want to work indoors, do as the pros do, soft single color background, soft lighting, no distracting objects unless it is interacted by the animal.
I could go on, but grab some books on portraiture photography, the basic rules apply in many cases, have a look at some really good work and analyze why it is so, and you will be well on the way
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