Making a bespoke wooden box.

[AlanBeckett]

The part of the drill press you refer to is called by two names; either "Spindle" or "Quill".

Col,

I would argue that the part that goes round is the Spindle and the part that goes up and down is the Quill. So, for example, Milling Machines with a fixed Vertical Head are referred to as having no Quill. At least, that's how I understand it over this side if the pond.
Our Colonial cousins seem to have different views - see http://en.wikipedia.org/wiki/Drill.

Alan