How do you convert a physical piece of artwork/picture into a digital one?
Depends whether the desired output is 2D or 3D. And what you want to use it for.
If two-dimensional, you photograph it and digitise the file into the format you require.
If three-dimensional, you convert the file into a hologram or set it up for CAD and output it from a 3D printer.
I'm going to add-on to Steve's answer.
The easiest way to digitize a file is to run it through a scanner. The end result (usually a .tif or .tiff file) can then be converted to other formats such as, PDF, JPEG, PNG or GIF. Doing a raw file conversion will sometimes been enough depending on the level of detail needed and what the artwork will be used for once digitized.
If your digitized file is blurry or pixelated if you are good at Photoshop or Illustrator (raster or vector) art manipulation try to clean it up yourself. If you aren't skilled in those areas look at using a freelance or in-house designer who will be able to clean up the file for you.
Keep your use in mind, as if you are using the end result in a printed work you need to think about having 300 to 600 dots per inch (DPI) or its metric equivalent for good output. Also the file will need to be created in CMYK (print) mode to ensure colors are accurate.
If you are using the file digitally, you only need between 72 to 150 DPI or its metric equivalent and the file should be created in RGB (screen) mode to ensure accurate digital colors.