Elemental bonuses affect stats of magic belonging to that element: overall damage (if attack magic) on all magic cast on an enemy target. Weapon Skill (if the WS has an element) and Skillchain damage are also supported by elemental bonuses. Summoner's avatars and by extension likely puppetmaster automatons and dragoon wyverns are not affected by these items.
While damage increases can be easily demonstrated, there has been no evidence to suggest any effect of day/weather on Magic Accuracy. Testing supports this: Even more on magic resist rates. Mr. Decay. Simple observation of the equipment menu also shows no effect of day/weather on Elemental Resistances from Enhancing Magic (Bar-spells).
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.