In Ultima Online's Animal Training system, Hit Point Regeneration (also referred to as "HPR" is an attribute that helps an animal recover hit points throughout the fight. Unlike the Mana Recovery formula which is very complex, the Hit Point Regeneration is very straight forward and has a linear progression.

The Hit point Regeneration value is the amount of Hit Points a pet will recover over ten seconds.  Hit Point Regeneration is Capped at 20 via Animal Training. 

The Animal Taming Mastery skill Consume Damage can give up to an additional 30 HPR to your tamed animal.
The Bard Peace Mastery skill aura Resilience can give up to an additional 22 HPR with 120 natural barding skills.

Below is an example of Hit Point Regeneration with max trained HPR and max consume activated:

Health Point Regeneration increases what is referred to as Effective Health. Effective Health is the time it would take a pet to go from full health to 0.

The longer the duration of combat, the more beneficial HPR becomes. At one minute of combat 20 HPR is equal to an extra 120 Hit Points on your pet. Coincidentally, 120 Hit Points and 20 HPR cost the same amount of training points (360). Any combat that extends over 1 minute and HPR becomes more valuable than Hit Points.

You typically always want to max the HPRstat on your animal. There are a few exceptions, such as when you are working on a tight spec on an animal and that already has over 550 hits (such as a Chiv/AI Bane Dragon). 

You can drop a couple points from HPR if you need to flesh out scrolls or intelligence, for example.