Feed shaft and Lead screw
Feed shaft  
The feed shaft (rod) is a power transmission mechanism used for precise linear movement of the carriage along the longitudinal axis of the lathe.  This is used to for automatic feed.  The engagement levers is situated on the apron

Lead screw

The lead screw of a lathe is used to advance the carriage of the lathe in time with the rotation of the spindle. It is used to make threads by having different gearing between the spindle and the drive for the lead screw. The half nut on the apron is used to engage the lead screw.