During battles, you may notice a bar on the right side of the screen. This bar is known as the Bonus Meter and once it is filled up, all battles until the gauge is broken become Bonus Battles. They basically are another aspect that can help you improve success in battles or gain something extra from the battle.

Bonus Battle Extras
Double Fol
Triple Exp
Increased chance for item
Increased recovery rate upon win

To fill up the meter, you just simply try to take as least damage as possible, make a lot of damage, create combos with your characters. Basically, dominate the battle and it'll fill up. But, once you are in Bonus Battle, too many taken hits may break the meter and you have to start all over again.