All mobs in the game are part of a set of families. Some families are more adept at killing certain other families. What the below charts mean is that, for example, Beasts are better at killing Lizards, which are better at killing Vermin, etc. You can see that it loops at the end.
As a Beastmaster, it is your job to learn these family weaknesses and exploit them by choosing a pet that's better at killing your chosen prey. Note that not all of the mobs below are charmable.
The effect of properly following this chart is intimidation. Intimidation causes your prey to lose a turn, so to speak, and it doesn't attack. Intimidation messages appear in your chat log as "The Steppe Hare is intimidated by the Strolling Sapling". It functions like paralysis, and stacks with any other paralysis effects (such as from the spell Paralyze or Ice Spikes). Beastmasters themselves eventually learn many of these "Killer Traits".
As a Blue Mage, it is your job to learn these family weaknesses and exploit them by using abilities learned from a family that's better at killing your chosen prey. For example, by using a spell learned from a mob of the Lizard Type against a mob of the Vermin Type, you can maximize your spell accuracy and damage. See Blue Magic Monster Correlation Chart. Also, by setting certain combinations of spells, you can obtain various killer effects.