Pets are friendly mobs whose primary purpose is to defend you. They are equippable, as is the case with most items.
When you tame a pet it's level will be lowered to yours. They will level up as you do, according to the XP that they get, so they might not be on the exact level as you when you level up. You can fight other animals that aren't on your level, but when you tame them they will either be on the same level as you or lower.
Pets will attack enemies that their owners have dealt the most damage to. For example, if one is fighting two separate enemies, the pet will attack the one who has taken the most damage from that player.
'Tank' and 'Healing' pets also are available to tame (see turtle and spitter.)
Pets favorite foods can be found as drops or in the general store. If they spawn inside cities, they are friendly; potential pets that spawn outside of cities can be either neutral or hostile.
They will be stored in the pet tab when not equipped. Pets can be called back to their owners using the T key. If in combat, the pet will stop fighting and return to its owner.
The stamina pets have is represented as water drops below at the bottom of their unit frame. Stamina is required to ride a pet and will deplete slowly while doing so, it can be refilled by letting the pet stand at any water source.
Pets that are too far/stuck can be teleported to the player by unequipping the pet from the 'Pet' slot and re-equipping it, or simply by pressing the T key to call them. Pets are not lost forever once they die, they simply re-spawn out of the player.
Naming a pet = press the return key (Enter) to bring up the chat and type: /namepet <name> (<name> being the desired name of the pet). If desired, to add spaces in a pet's name, press the Spacebar then hold Ctrl and press the Backspace key. The game thinks there is no longer a space there, but there is, allowing one to give a pet a longer name like 'Terry the Great'. This method may be changed in future updates.
Taming
Pets are tamed by holding the associated pet food out for them to eat. Simply equip the food item into the pet slot and go up to the pet desired. The pet does not have to be friendly to tame it. Just equip the pet food before you are in sight of the harmful pet and then approach.
If more than one neutral animal is trying to eat your pet food, the one which finishes first will become your pet, while the others will start attacking you.
Bosses can also be tamed, when a boss is tamed it will change into normal sized pet, and sometimes will duplicate the boss in the world, causing it to continue attacking the player.
Pet Types
Melee pets are the most common pets and attack from close range.
Ranged pets can shoot projectiles from a distance.
Tank pets deal less damage but have higher endurance and generate more aggro.
Healer pets have healing abilities to heal their master and friends.
Some pets are ridable and enable faster traveling.
Leveling
Pets gain experience when Mobs are killed (but your pet needs to be alive when the mob is killed to gain the experience). A pet will receive the same experience as their master receives. This appears to be broken sometimes while playing in parties. Pets keep their level and experience when stored.
Trivia
Wollay also stated that he plans on adding 'Pet Evolution', which will enable for certain pets to evolve overtime.
Wollay has mentioned the addition of Rare Pets with special looks and abilities.
Rare Mobs take longer to tame.
Riding on water does not deplete pet stamina.
Riding Skill also affects pets' swimming speed when riding.
Bugs
When on a server, pet items are not updated when a pet earns experience or gains a level. So when playing on a server, pets will lose any experience or level gained whenever one changes a pet, or due to a bug whereby a pet seems to be unequipped and reequipped.
However, pet items are updated correctly in single player mode. So training one's pets in single player will work, and this will carry over to multiplayer mode.
Stunned pets are mountable.
If you attack a target dummy or a scarecrow, your pet might keep running towards it regardless of how far away from it you are, until the pet is killed and re-spawns. You can recall your pet by pressing 'T'.
Pets killed while riding will still give movement speed to player. Characters will float giving the impression(animation) that it's still riding a pet.
If you unequip and reequip your pet, he will regain all his HP (even if he was dead)