Whenever a Pet gets 99.99% XP and is taken taken to a Pet Tamer and you complete his Quest with your Pet out, it will Level Up (Which will give your Pet more HP) and it's Rank will go up.
When your Pet gets its Rank up, it will give more of the Stat that it gives.
Example: a Lion will give more STA, a Unicorn gives more HP, ect.
The amount of Stats your Pet will give while using it will vary depending how high the Bonus is.
There are 5 Ranks, D, C, B, A, and S.
When an Egg hatches into any Pet, it will start out at Rank D. At rank D, it will always have the Rank Bonus of 1lv. When it levels up, and it's Rank goes up, it will be able to give more depending on what Rank it levels up into, and of course by chance. The Rank bonuses increase in jumps of 2 lv.s.