Fina asked us this today when setting up an EX+ team for the upcoming Fire Guild War, and, thinking about it, it isn't necessarily intuitive:
When Awakening a character, you get to pick from four different types of Awakenings to apply. All characters start in Balanced, and can be changed to the other three types by consuming a corresponding Awakening Orb.
Important: Awakening Orbs are a limited resource. If you don't feel the need to change awakening, it's best not to spend them just for the sake of doing so. Standard is generally okay, and switching is usually something you do based on need to optimize further.
With that said, let's have a look at our options:
As the name implies, Attack maximizes the character's... Attack. In practice, what this actually means is somewhat nebulous - this means the Raw Damage value. If you haven't read it yet, please check out my breakdown on Damage vs Damage Cap to understand what this mens.
C.A. DMG is also modified, but not C.A. DMG Cap, and since this type already increases raw damage so much, the C.A. DMG upgrades are mostly inconsequential. Attack Awakening also loses out on the HP and MA buffs from Standard.
Attack is best taken to reach a high damage cap you otherwise can't hit. This is mostly relevant for characters with Assassin modifiers, as without this extra juice, it can be tough to bring out their full potential.
As Attack provides no Damage Cap benefits, if you're already capping, this may result in you just throwing away your HP and MA benefits from Standard for no reason, so use with caution.
Fortunately, Defense is a lot more straightforward. It increases your Defense and HP, at the cost of losing out on most of the offensive buffing Standard provides.
That's about it. There really isn't much else to say.
Defense is the least general of the three types. Remember the common adage of "The best Defense is a good Offense" - as in, if you kill the boss faster, the boss has fewer chances to hurt you.