There are numerous monsters all over the world of Vana'diel. The vast majority of monsters belong to a certain Family of monsters. A family of monsters will generally possess similar traits and skills. For example, both the Beach Bunny and Variable Hare belong to the Rabbit Family and will typically possess the same abilities (e.g. Double Attack) and special attacks (e.g. Foot Kick). In addition, monsters from within a family will generally share certain behaviors and weaknesses (see Monster Classification).
Each family also has a Type, which generally describes a broad class of monster. Thus, the Rabbit Family is of the Beasts type. Monsters of one type may possess certain advantages or disadvantages when interacting with monsters of another type. To see how types interact with one another, see Ecosystems.
All monsters in the game are part of a set of families. Some families are more adept at killing other families. For example, Beasts are better at killing Lizards, which are better at killing Vermin, and so on. 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's presence" (But the enemy must be attacking the pet that has this trait to be intimidated- "presence" isn't good enough). 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.