Invincibility plays a key role during the end-game of Kingdom Hearts II: Final Mix. It is basically a 'helping hand' when trying to dodge attacks. You don't physically 'dodge' something with the "Dodge Roll" Ability, instead you are invincible while rolling around. There are many ways to guard/dodge incoming damage, here are a few examples.
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A Dodge Roll makes you briefly invincible during its animation. This can be used to dodge strong attacks easily, like Lexaeus 'storm-attack', where he first dashes towards you and then makes the whole ground literally explode with rocks.
You can dodge nearly every attack in the game by dodge-rolling at the exact time of damage. The few abilities that can't be dodged are for example the Grab-Attack by Lexaeus, where he first turns 360° and then grabs you. This specific attack can be dodged by either jumping far away or reflecting it with Reflega.
Using Guard at the exact time of an attack can block all incoming damage and may make the enemy stagger into an vulnerable phase of weakness. You also get a few moments of invincibility afterwards, if you block the incoming attack correctly. This can be shown with Terras Slice-Combo at the beginning of his Ultimate Move. I blocked 2 attacks of his dozens of swings. At the last downward swing I just stood still and he didn't hit me. This is because of the earlier Guard that made me briefly invincible. You can block every sword attack with a good timed Guard.
Cure itself doesn't make you invincible, but the Ability "Leaf Bracer" does. "Casting Cure on yourself will continue even when attacked". It basically makes you untargetable during the moment you hit the button until your HP got restored.
"Once More" is one of the most important skills. It lets you survive a devastating combo-attack with atleast 1 HP,but only if you have more than 1 HP before being hit by the Combo. Pretty self-explaining. The Same goes for "Second Chance", which lets you survive a single blow with min. 1 HP.
Using the "Aerial Recovery" can block incoming damage if timed correctly. The same goes for the "Aerial Dodge" ability, the ability you get from "Master Form". If timed correctly, both can guard you from incoming damage or even reflect projectiles like Terra's Cannonball.
Specific Limits like Donalds "Duck Flare" give you invincibility while still being able to play and attack normally. This is extremely helpful during some of the long-lasting attacks like Vexen's Ice Barrage.
Going into a Drive Form makes you briefly invincible until you finished 'loading' into the Form itself. The same goes for the Limit Form Limits: You are invincible during the first few seconds of any Limit from the Limit Form. You can use that to push through attacks of your opponent, for example Roxas' Combos.
Some People might not know this, but you deflect/guard nearly everything while jumping in Final Form. Many people complain that Final Form is bad because you barely have any quick mobility or dodge-abilities, but not many know that you can just jump your way out of any incoming damage you might've received.