On this page of the Disgaea RPG guide, you will learn what happens when your character or the whole party dies.
Do not worry if your character or even the whole party dies during a battle. You are in the underworld! After losing a fight, you will find your characters safe and sound on the Character List in the Party tab. You can immediately use them in the next battle.
When one or more characters die but you win the battle, all characters, even the fallen ones, will gain experience points.