Taming is the process of making a monster fight for you. When any monster is defeated, you can attempt to tame it and keep it in your team- in Pokemon, which Monster Ranch is extremely similar to, this would be noted as "Catching" a monster. They will never leave your ranch, where all monsters go once tamed, unless you release them, trade them with another player, or add them to your party.
If a monster is rare, has a high level, or both, you may have to use items. These items increase the chance of the taming the monster. You can see what chances you have of successfully adding the monster to your ranch by the messages it gives when you click "Tame this monster". Fortunately, you can buy items from the General Store in the Marketplace that will increase your chances of catching it.