Pivot cups are the sleeve of where the hangar meets the baseplate.
Pivot cups are generally very specific to each different truck. Some trucks can take a standard "Small" Pivot like the Khiro Small Pivot Cups, or similar to the Riot Small Pivot Cups. In the description of the Riot Cups it will always list what trucks work with what, and if you still have any questions as far as fit goes don't be afraid to email or give us a call at the shop.
As far as the soft or hard, or the durometers of the pivot cups, soft cups will give a more smooth/ dampened feel when leaning and turning, but harder ones are more responsive with a quicker turn feeling. Usually the hard pivot cups will also last a little longer than the soft, because of deformation of the softer cups.