Simply put, it's a boost to psynergy, summons and element-based weapons of that element for the remainder of that battle. This answer can be found on this page and in this thread and in Krazer's comment below.
From the Golden Sun: Dark Dawn Manual under the section Battles (page 30 in my copy):
This is the character's strength for each element. The higher the value, the greater the character's attack strength with that element.
When selecting a psynergy target, in the lower right corner, you will see the elemental power indicated. You can also check this value in your detailed status screen. Note that the increases due to djinn summoning will only last until the end of that battle.