打仗 (dǎzhàng) is a separable verb (离合词) meaning 'to fight a battle / go to war' — the 打 part changes (打了, 打过) while 仗 stays as the object. It's the everyday word for war or battle. 战斗 can be a noun or verb and refers to any fight or struggle (not just war); 作战 is a formal, military-only term for combat operations.