Every card has a number value rangin from 0 to 9 printed in the corner. When you use that card, it remains in play while its action is being fulfilled (perfroming an attack,casting magic,using an item, summoning things to assist you, etcetera).If an enemy uses a card that has a higher number value than the card you are currently using, the effect of your card is negated and the card is 'broken'. Such an event is reffered to as a 'Card Break'. The same situation applies if you are to use a card with a higher number than that of your enemy, in which case, his card os broken
That's all that I can be bothered explaining, but that's probably the most important part of understanding the battle system.
Someone else please continue from where I left off.
Cards with a value of 0, if played after an enemies card, will break any value of card. If the 0 card is played first though, anything can break it.
Sometimes a fancy 0 card with a green Mickey Mouse symbol on it (someone correct the name), which, when played, will send the enemy (boss battles only) into a spiralling unconsciousnessnessness for a few seconds, leaving them vulnerable.
You can stock 3 cards to preform sleights. Sleights are like Ragnarok from KH1, except far more plentiful, and need a certain card combination to be performed. You may need a cartain card value, or certain types of cards to execute them. The first card used in a sleight is wiped from your deck for the remainder of the battle.
You can power up your magic and Summons by stocking 2 or 3 of the same type of card, eg. stock 2 cure cards to perform cura, and 3 cure cards to perform curaga. Summons just have better attacks with more cards stocked. The first card used in one of these combinations is also removed from the deck for the rest of the battle.
In most battles (all?), friend cards will drop from nowhere in particular. Pick them up to add them to your deck. Friend cards work like summons. Stock them for stronger attacks.
Sometimes enemies drop enemy cards. You can store these in a different part of your deck, and they can perform special abilities (like restore HP after defeated once) when used. When an enemy card is in place, it can stay active for a certain amount of attacks, or just go away after a single use etc. When an enemy card is used, it is removed from the special deck and cannot be used for the remainder of the battle. Playing another enemy card while another is still in place cancels the effects of the first enemy card played.
There's not much else to it, lest I'm forgetting something. Someone can add anything to this if they want.