Enchanting allows melee attacks to take advantage of elemental weaknesses. However, there is more to it than that.
Enemy Weakness
The enemy's weakness will be multiplied by a value depending on whether the enchant is single element, or combined. While combined element enchants such as explosion or ice can effectively hit two weaknesses, the weakness is reduced slightly (multiplied by 0.9). Take for example a monster with a -100% weakness to water. With a water enchant, they would take double damage. With ice or lightning enchant, your damage output would be multiplied by 1.9. Racial damage bonuses and elemental damage bonuses also factor in synergistically.
Elemental Damage Bonuses
With an enchanted weapon, elemental bonuses also factor into your damage output. This isn't necessarily your elemental bonus, however. In the case of casting enchant, it is the caster's elemental bonus at the time of usage that applies to the enchanted weapon. If you are using an elemental weapon or cast enchant on yourself, then your elemental bonuses are factored in. Enchant I applies elemental bonus at a rate of 0.4, versus 0.8 with Enchant II. For example:
30% water elemental bonus with Enchant Water I: 30% * 0.4 = 12%, damage output multiplied by 1.12
30% water elemental bonus with Enchant Water II: 30% * 0.8 = 24%, damage output multiplied by 1.24
While this itself may not sound significant, the bonus here is multiplied by the enemy's weakness. Basically, a high elemental bonus will further exploit the weak points of enemies.
40% water bonus, Enchant Water I, 50% water weakness: 1+(40% * 0.4) * 1+(50% * 1) = 1.725 damage multiplier
40% water bonus, Enchant Ice/Lightning I, 50% water weakness: 1+(40% * 0.4) * 1+(50% * 0.9) = 1.682 damage multiplier
40% water bonus, Enchant Water II, 50% water weakness: 1+(40% * 0.8) * 1+(50% * 1) = 1.98 damage multiplier
50% water bonus, Enchant Water II, 100% water weakness: 1+(50% * 0.8) * 1+(100% * 1) = 2.8 damage multiplier
Combined Damage Type
If a condition is met there is a fundamental change to the attack strength and damage type of your physical attacks. This condition is…
(pATK + mATK)/2 * 1.1 > pATK
In other words if the average of your physical and magic attack times 1.1 is greater than your physical attack alone, while enchanted your attack power becomes the average of your physical and magic attack multiplied by 1.1. In addition, your normal attack as well as any skills that were physical attack-based
will become attacks of combined physical and magic 
. What this means is that instead of the enemy's physical defense, the average of their physical and magic defense will be taken into account when you strike them. Taking advantage of this, it is possible for melee types to defeat elementals, who have SS class physical defense.
