What is the name of this tool that is used to clamp a piece of wood for drilling?
It's a G-Clamp (or C-Clamp, or F-Clamp, or whatever other kind of clamp works) with a piece of scrap wood.
The clamp clamps the scrap wood to the back of your workpiece so that, when the drill bit pops through, it doesn't break the back of the workpiece but instead drills a neat exit hole.