Taming is a Fauna skill used for acquiring pets. The pets can range in purpose from attacking your enemies, give increases to your combat skills and/or land skills (plains/forrest/hills/mountains/desert/cave) or just looking pretty for aesthetics value. Pets can also be used to help in more specialised skills. For example, botany (boar) or extracting (varanus). As such, Taming is considered a very valuable skill especially to aid in dungeons throughout the game at all stages.
After the patch v1.17, taming can now be done in a building called bestarium.
Assuming you have 100% energy, you can do any of the below tasks 6 times. Earning a total of 4.8 gold, 8.4 gold or 24 gold respectively. Paying a small amount to sleep in an inn or building your own home will provide you with comfort which in turn will provide bonus Energy when you sleep. Increasing the amount of tasks you can do and thus increase your potential gold income. These are repeatable tasks.
This task requires 20% HP. Meaning it will damage you for 20% health to a random part of your character each time without fail as if it was a monster. You can still do the task mutilated, but you risk failing it if it hits the mutliated part.