I thought a bit about this and did a little testing on this, and I came to these conclusions (feel free to correct me if I'm wrong):
1) Elemental Resistance Against Physical Attacks:
If an enemy is resistant to your elemental based physical attack, you will deal 1/2 as much damage as normal... or in the worst case, Dark Knight (Darkness) weaponry vs Zombies (is there a specific multiplier/divider?).
If your weapon's elemental is effective against the enemy, you will deal 2x damage (sometimes 4x, unless there's something specific I'm missing here?).
If your weapon has an enemy type property, a specific "enemy type multiplier" is in use? In other words, all anti-Dragon weaponry will deal 4x damage, regardless (Whip, Spear, Arrows)...
If you are two handed (Yang/Edge), or your weapon has both an elemental+enemy type attribute... the target that has a weakness to the any of the attacks will take precedence over the resistance (if it even applies).
For instance, if you attack a FlameDog with a FireClaw+IceClaw, the Ice Claw takes precedence over the Fire Claw, therefore dealing awesome damage.
In FF4A, the Apollo Harp has the Fire attribute (I think) and the Dragon attribute. Attacking the Blue Dragon will allow the Dragon attibute to take precedence over the Fire resistance. Similar thing would occur with the Dragon Claw+any of the elemental Claws...
2) Elemental Resistance Against Magical Attacks:
If you are weak to an elemental, you will take 2x (or 4x in some instances?) damage.
If you are resistant to an elemental, you will take 1/2 damage (rare for enemies, usually applicable to character equipment).
If you can absorb an elemental, invert damage intake (though, some attacks seem to do 1 point of healing).
If there is an uninterrupted sequence of attacks, any elemental based attack that you can absorb or resist (absorbing taking precedence), all further attacks will accumulate against applied absorption/resistance.
Does this sound about right?