On a mens dress shirt the placket is the thick strip of fabric that contains the slits which act as receptacles for the buttons on the front of the garment. The purpose of the placket is to reinforce the fabric around the high traffic area where rips and tears are most likely to happen. When you consider the amount of force and fabric stretching typically involved when buttoning and unbuttoning a shirt it is easy to understand why this extra reinforcement is necessary.
The term placket comes from the French word plaque which translates to "to lay on". This translation makes more sense when you consider that the term placket is also used to describe strips of fabric used to conceal openings on garments; such as the small flaps of fabric typically found over zippers.