Quality indicates the stats (health, attack, and speed) of a pet. The higher the quality (rare being the highest at the moment), the better the pet's stats will be and they will scale even better as they level.
Poor (grey) is the lowest quality, and as quality increases from white to green to blue, so do their health, attack and speed values. I'm not sure about the exact value differences between qualities, though.
So for example, let's say I have a poor (grey) quality cat that's level 3. My friend has a rare (blue) quality cat, and it's level 3 too. Their pet's health, attack and speed values will likely be higher than my own, even though both are the same pet and level. This gives them an advantage over my own cat while in a battle, since speed will determine who goes first, and more health and attack power is just good in general.
Stats only really matter if you intend to battle with your pets. If you enjoy having a companion just to have it, then by all means, you can ignore its quality.