The Critical Element Skill will allow Elemental Damage to strike critical hits. The ingame Monster Field Guide contains information on Monster weaknesses. Just because a monster uses fire does not mean it is weak to water. Some important things to note when considering Elemental Damage against Monsters Armor with high resistances to certain elements will cause you to take less damage and reduce the effects of Elemental Blights.
Monsters can also deal elemental damage to the hunter. For example a fire monster may take more elemental damage from weapons that deal water damage but less or zero elemental damage from weapons that deal fire elemental damage.
Each monster has weaknesses and resistances that will cause it to take more or less damage from specific sources.