You can tame certain monsters when they are in a weakened state during combat. Once taming is triggered, you need to approach the monster and long press (A). You will then be prompted to do a Quick Time Event which you need to succeed in order to tame the monster. Tamed monsters are transferred to the estate lobby.
Figure 1: Weakened State
Once you have tamed the monster successfully, you can check it out in the animal pen inside your estate.
Some tamed monsters can be trained as a mount. You can view which monsters can be trained as mounts by checking the stables tab. Other tamed monsters can be trained inside the animal pen as livestock to produce materials that you can gather.
Here’s the list of the tamable monster: