Pet's main feature is the movement speed bonus. This bonus is only active on open world. Some pets also have their own level and skills which can be really useful.
Eggs are the items you equip to spawn the pet. You can see pet's experience, speed bonus and skills. You can get eggs from dungeon prize chests or quests.
Pets automatically cast their skills when there is a target. Some skills get stronger or gain bonus effects at higher levels.