ATK Power determines the ATK (attack) deck cost you can support. When creating an attack deck, your deck cost cannot exceed your current maximum ATK Power Points. You can however create ATK Decks that are much lower than your maximum ATK Power but all decks must be a minimum of 3 cards. Since all cards have a deck cost of at least 2, the lowest cost deck costs 6 ATK Power.
During PvP (Person vs Person) Battles you will select one of the 4 ATK decks to use against your opponent's DEF deck. Certain offensive Skills can give boosts to all other cards in your ATK deck while others only modify itself. In PvP Battles, only your overall modified ATK total vs the overall DEF total of your opponent will determine the winner. There are 2 exceptions to this rule, one involves Mimic Traps, the other is when a Judgement of Shame is triggered.
In raid or quest boss fights, the defense values of your ATK Deck do play a role in the outcome of the battle. In these battles each side exchanges hits until one side's defense total is brought completely down to zero.
You can create 3 separate decks, while a 4th Recommended Deck is automatically created for you. The Recommended deck will make sense early on as it will keep adjusting your attack deck to maximize your attack potential, but later one, you will want to use your own pre-built decks for different situations. For example there is no point in using your most powerful deck against a weak opponent as you would be going overboard and waste ATK Power which takes time to recharge. Also the 'Recomended' deck does not always take into account card Skills or Battle Formation bonuses that you might want to use againsts certain opponents.