That's for when you get the Barrier Materia. When you use the different barrier spells (the number depends on how high the materia's level is) this grid shows how long your character or characters are protected.
The Barrier materia has the following spells:
Barrier
M-Barrier
Reflect
Barrier & Wall
Each of these allows the user to cast a protective shell of sort around them and the little bar beside your name is the amount of power left that you have before the barrier collapses so to speak. Then you'll have to cast a new one. It's highly effected on the tougher bosses.