Controlling Protoneer CNC shield without PC
An Arduino can talk directly to a shield, independent of a PC.

This will, however, need a way to tell your Arduino to do it's thing. Depending on what you want it to do, this may require nothing ("send this pre-defined g-code as soon as you are turned on"), a memory card of some sort, one or more pushbuttons, or even a small screen. I have seen several 3d printers with memory cards, buttons (usually up/down/select/back), and a small screen that, once set up, can print anything you can load onto a memory card.

There are other forms of communication, too - ethernet, wifi, etc., which you can use, with appropriate shields.

In the end, of course, you still have to design your object - 3d graphics are well beyond the ability of an Arduino, so there will still be some form of computer involved.