Progressing through the game will eventually allow you to learn the skill to tame pets.
To tame monsters to be your pet, you need to be level 45 and you first purchase bait from any of the Pet Supply Merchants.
In the Codex you can find useful information about the monster you wish to tame, such as the minimum Charisma your character has to have and which bait you need to tame the pet.
Your character may not be able to summon some pets, if your character's Charisma is not high enough.
Also note that some monsters cannot be tamed.
※ Depending on your character's class(job) or play style, the moment at which you learn pet skills may differ.
※ Archers have the ability to tame a pet near the start of their journey.
When summoned, pets will assist in combat, and their skills or abilities will be enhanced as their XP increases.