Your party members can't just get by on normal attacks alone. Each character has access to plenty of skills, not just unique to each character, but also from their weapons as well. Since monsters have their own weapons, which doesn't have their own skills, they get their own skills upon leveling up.
Skills sometimes have an element attached to it, which can deal more/less damage to an enemy. This is because enemies sometimes have an elemental weakness/strength that will limit or increase the damage they take. There's five different elements in the world of Disgaea, which are Fire, Ice, Wind, Star and Non-Elemental. Star doesn't have any innate resistance, except for the magic resistance stat.
