ATK (Commonly referred to as Attack Points or even Offense Points), are the points that determine a Monster Card's "strength" when attacking the opponent's Monster Cards.
If one ATK is lower than the other, the one with the lowest ATK is destroyed. The owner of the destroyed Monster takes damage equal to the difference between the two ATK values, unless a card effect prevents Battle Damage.
If both ATK values are the same, both monsters are destroyed (called a Double KO), and no damage to Life Points is inflicted on either player, unless a card effect dictates otherwise.
If both the ATK and DEF values are the same, neither monster is destroyed, and neither player loses Life Points.
If the ATK value is higher than the DEF value, the Defense Position monster is destroyed, and no Battle Damage is taken, unless the attacking player has a Piercing effect that deals the difference between the ATK and DEF values as Battle Damage.
If the opponent has no Monsters on the field, damage is dealt to the opponent's Life Points equal to the total ATK value, if the attack is not blocked or disrupted. This is called a Direct Attack.