This came with my lathe kit what is it for etc?

Chris,

It's a balance chuck. The balance goes inside with one pivot of the staff protuding from a small hole on the front. The plunger from the back is moved forward and the other pivot is carefully placed in the hollow tapered end. The plunger is set into place with the lever on the side. Now you can safely polish the pivots of the balance while the arms of the balance are protected inside. Nice thing to have.

ps. remove the hairspring before attempting this

Thanks Mike, Simple once explained and may be a useful tool on occassion.