What does the button next to the Pivot Point button do? In the 3D view port's header, what does the button next to the Pivot Point button do? The button in question has a shortcut of alt ,. It is in the red square in the picture below. 


What is it use for and how is it used?
That button  is the manipulate center points option, which allows you to manipulate the origins of objects without affecting the rotation and scale of the objects themselves.

For example, if I scale these objects without it enabled, they are stretched. With it enabled, only the origins are moved (only affecting object location):



Note that with only one object selected, enabling this option will cause scaling or rotating the object to do nothing.