A pet has 8 stats.
Health is the most straight forward of the stats. When a pet's current health reaches 0, they die. Higher health is always ideal, as it determines whether certain items can be used. It also determines how much they are healed from potions and supports, both which heal based on the target's max hp and other factors.
Attack determines the base damage a pet does, modified by the attack's element and the opponents element and defense.
Defense determines how much damage is reduced from an enemies attack.
Speed players determines 3 things. The first is the turn order that every enemy and pet acts. The second is the amount of action a pet has in a turn. Lastly, it determines how much extra damage a pet does if it has more speed.
The turn order is pretty self-explanitory. If a pet has higher speed, the odds of it acting first increase. A pet with higher speed will not always act first. Having a slow party means pets will likely take damage before even striking. Having a fast party can result in dealing damage before the enemies get even a chance to act.
Pets can gain more than 1 action per turn depending on their speed. Each speed gives a pet 0.2% chance to perform an extra action, with pets having more than 500 speed being guranteed a second action. Each additional 500 speed will let the pet act an additional time. More actions mean either more damage or healing, so having speed is important for your damage dealers and healers.
Lastly, pets gain additional defense-ignoring damage based on their speed. A pet with greater speed will deal half the difference in defense-ignoring damage.
One thing to note is that enemies also benefit from higher speed. They will act sooner, more often, and deal defense ignoring damage. This is important as this fix damage can result in a dead defender if they don't have the health to cover it.
Another thing to note is that in depth 3, enemies always act first on turn 1. Past turn 1, speed becomes relevant again. This means that the value of speed drops later on, as defense and health become important to surviving this first attack.
Fire, Water, Earth, and Wind
Each stat affects how much elemental damage is taken and the chance of being hit with a status effect from each element's respective monster.