There are four elemental crystals: Fire, Ice, Wind, and Lightning.

The crystal provide extra damage and defense.

Fire Beat Wind (F>W)
Wind Beat Lightning (W>L)
Lightning Beat Ice (L>I)
Ice Beat Fire (I>F)

If a character is equipped with a fire crystal, the character will do extra and take less damage to monsters with wind crystal. It is possible to one shot a monster or get one shotted by a monsters with the crystal mechanic. Keep this in mind when planning how you want to engage the monsters.

Some characters have skill(s) affinity to certain elements, so it best to pair the correct elemental crystal to the character. For example: Noir have a skill called Volcano Drive which is firebase. So fire crystal should be equipped for Noir.

Noir = Fire
Vert = Wind
Neptune = Lightning
Blanc = Ice

Elemental Treasure Chest can only be open by element that is weak against. In order to open an Ice chest, you need a character equip with a fire crystal.

Buy better crystals when it becomes available.

Mastering the crystal mechanics is very important near the end of the game. There are bosses/monster with AOE that can one shot your character(s) with the wrong crystal. You need to use characters with the correct crystal to kill the monsters before half your team gets one shotted.

To find the enemy element affinity. View the enemy and see which elemental crystal the monster have. If monster have no crystal the monster is neutral.

Buffing a character and/or debuffing a monster/boss + using the correct cyrstal can one or two shot the monster/boss.

To further increase the damage use the Lilly Boost mechanic. Lilly Boost is where ally characters are placed adjacent to one another during battle. The maximum lilly boost is four (north, south, east, west) with the attacking character in the center. With max lilly boost, buff, debuff (monster) and the correct crystal equip character it is possible to one shot a boss or very tough monsters. In this format damage can exceed up to or greater than 10k.

If possible use an elemental skill over a special move. The element skill with the correct crystal equip will do much more damage than the special move. For example: Noir equipped with a fire crystal will do much more damage with Volcano Drive than with Special Move against a monster with affinity to Wind

There are some levels where monsters use one element. In this case it is better to equip the counter crystal for the entire team. For example a level where all the monsters use lightning so the entire team should equipped with Wind crystals.