On your quest, you will encounter enemies and you will have to engage in a battle with them.
The screen will change to the battle screen when you encounter an enemy.
The battles in Fool'sQuest are turn based battles, which means you choose to preform an attack or use an item and the fastest participant in the fight (be it foe or ally) starts first. Then each character will takes turns.
The first slot(here it is named "Kerli") is the slot for weapons equiped, so Marina fights using a Kerli Weapon in the above screenshot.