Elemental bonuses affect stats of magic belonging to that element: overall damage (if attack magic) and Magic Accuracy on all magic cast on an enemy target, in addition to elemental damage and elemental resistance created by Enhancing Magic. Weapon Skill (if the WS has an element) and Skillchain damage are also supported by elemental bonuses.
Usually, these bonuses are randomized to some extent (possibly managed by Moon Phase, though no one is exactly certain). This attribute presumably ensures that all of the listed day/weather's bonuses for the wearer will be at the maximum when the item is equipped.
The attribute does not appear to govern elemental bonuses obtained from other pieces of equipment affected by elemental days or weather; Only base player statistics appear to be affected.
It is currently unknown as to whether this attribute can apply to Synthesis results.