To belay facing the wall while suspended by your harness to anchors. Hanging belays are done when there is no suitable ledge or foothold available. Most often done on difficult, multi-pitch climbs.
To belay while suspended by your harness to anchors without a substantial ledge to stand/sit on. Atmospheric.
A belay point which you to hang suspended, feet not touching the rock at all. This is usually when your harness starts to bind, you need to go to the bathroom, and you become convinced that your anchors are slowly wiggling their way out of the rock.