King is correct about the offset stuff -- but there's more info.
A) It's (usually) called a "cup center".
B) The reason for the offset is this: if you turn a piece with one end centered (in the "live center", so named because it's attached to the motor and thus directly driven), and the other end offset (in the "dead center" -- in this case, your mystery part) -- the resulting piece, when turned 'normally' (i.e., as if it's a straight cylinder) will come out
*tapered*. Lathe attachment that offsets the tail stock as you rotate it, to let
> you turn things like table/chair legs with one end offset from the
> centerline. The more rotation the more offset.
> Si
>
> Michael Sorbera wrote:
>> [quoted text muted]