It's a lathe dog. It grips the part and extends over so it can be pushed by the spindle so the lathe motor can turn it. Like the chuck, it grips the part so the part must follow it in the circumferential direction but unlike the chuck it does not impose any radial positioning to the part.
Why is this useful? Why not just use the chuck? Well, a chuck grips the workpiece from the outside and does so relative to and symmetrically around the turning axis of the lathe. This will inherently assign a central axis around which the part will turn. That's bad if you need a particular central axis but can't trust the roundness of your workpiece where the chuck would grip it.
What you do then is put centers (conical holes) on each end of your part and mount it on the lathe using the centers. Now the turning axis is whatever you defined it to be independent of the outer shape, but now the chuck can't grip it to turn it. That's where the dog comes in.