Dead centers are almost always used on cylindrical grinders. The one that is driven is called a live center, the one in tailstock is called dead center.
On a lathe, Dead centers are useful for putting in spindle bore taper for between center turning, In the tailstock you use a Bearing Center or a lubricated dead center at relative slow speeds. And as with a grinder, the center that driven or rotates with spindle is called a live center. Regardless of what tool sellers call them, unless it is driven with machine power, it is NOT a live center. Should add that carbide tipped dead centers are a must for super precision turning or grinding.