Taming is one of the many ways a player can get a pet. There are however multiple skills that allow you to tame a monster as a pet. They are predominantly class based skills, to get them naturally you must be the right class however you can find items with bonuses to them which will grant you the ability to tame.
Beastmastery, this skill is gained by being an Archer that evolves into the Beastlord or Beastmaster paths, it allows you to tame Beasts such as Cows, Cats and Dogs and anything else that falls into the BEAST classification.
Demonology, this skill is gained by being a Caster that evolves into a Warlock, it allows you to tame DEMON classification monsters such as Lesser Daemons and Daemons.
Druid Magic, this skill is gained by buying the Hermit bought profession then evolving into a Druid, it allows you to tame Plant classification monsters such as Munchers.
Necromancy, this skill is gained by being a Caster that evolves into a Necromancer, it allows you to tame UNDEAD classification monsters such as Vampires and Skeletons.
Slavemastery, this skill is gained by being a Fighter that evolves into a Warlord, it allows you to tame HUMANOID classification monsters such as Humans and Elves.
There is also the Myths and Legends and Magical Mastery (not implemented yet) skills which allow you to tame MYTHICAL monsters and MAGICAL monsters, these only come as bonuses on items and are not profession based.
Aside from using Taming skills, you can buy pets from other players or where possible summon pets.
Check the Marketeer at 31, 59 after leaving Springlade to see what pets others are selling, you do not need any skill to own a pet.