What DPI should I resize my image to for best printing quality?

And my apologies if this sounds a bit too basic but I can't get my head around this.

I have a digital image I took with my camera. 4000x3000 pixels, and GIMP claims that its resolution is 72x72 DPI.

I would like to print a thumbnail of this picture in the highest quality possible. What I was told is that the printer which is going to be used works optimally with images set for 300 DPI.

In the printed document, I would like my image to be exactly 166 pixels wide, or 3.32cm (as Microsoft Word sets it).

And now to the question: how do I calculate how to resize my 12MP image, so when I take that image and import it into my word processor, it will end up printing the best?

My initial thought was this: Since the image on paper is going to end up being 3.32cm wide (1.31"), I should resize my image to be 300x1.31 = 393 pixels wide and set its DPI headers to note "300x300".

However, I'm pretty clueless about photography in general so I'm afraid this sounds as if I'm smoking something cheap.

Am I missing anything?

Answer

When it comes to print, terms like DPI, resolution, PPI, etc. get thrown around without much care or concern as to what they truly mean. So, before I send you off to a more in-depth answer about DPI, PPI, resolution, and print, a quick summary:

• DPI: Dots Per Inch
  
  
  
  • A 'dot' is a single element of a pixel
  
  
  • On a computer screen, a dot is a single 'sub-pixel' element, and may be red, green, or blue
  
  
  • On a print, a dot is a single droplet of ink expelled by the print head
  
  
  
  


• PPI: Pixels Per Inch
  
  
  
  • A 'pixel' is the smallest element of an image, "PIcture ELement"
  
  
  
  • On a computer screen, every pixel is composed of three 'dots' or sub-pixels, one red, one green, one blue
  
  
  • On an ink jet print, every pixel is composed of numerous dots of varying ink colors, usually a mix of cyan, magenta, yellow, and black, although modern printers often have several other inks as well
  
  
  • On a dye sublimation print, every pixel is a single dot from a blend of varying in colors, such as cyan, magenta, yellow, and black.
  
  
  
  


• Resolution: Variable meaning
  
  
  
  • Regarding an image, the resolution usually means the width and height of an image in pixels
  
  
  • Regarding a print, the resolution usually means the number of pixels in an inch (or cm, if you are from a country that uses metric.)
  
  
  • Regarding a computer screen, resolution usually means the width and height of the screen in pixels, but can also mean pixels per inch (i.e. 72ppi is the common "resolution" of the average LCD screen, while higher end screens often have a resolution of 100ppi.)
  
  
  
  
  

To answer the rest of your question, I've written up an extensive article here on Photo-SE that answers that question in great detail: