Printing is a pretty tricky process. The computer has to take the digital files and pixel placement and translate it into coordinates for a little desktop printer to deposit ink in lines as it travels at about a foot a second (in the case of ink-jet printers). Its a complicated process, but has become so refined that amazing results can be achieved.
There is one drawback with this system. Color. Our eyes receive light and red, green, and blue wavelengths of light are interpreted by our brains to produce an image. This is convenient for us because we are able to produce red green and blue light quite easily. Our TVs, phones, computers are all using this same color system, and they look great. They produce these colors which add to form new colors. This is called an additive color system.
Print however operates on a different color model. It is called a subtractive color system. It involves Cyan, Magenta, Yellow, and Black. When colors are printed on a page, they subtract the light wavelengths that are reflected to our eyes, not adding colors together to produce light. In this system, red and green don't produce yellow, for example, but produce an orangish color.
The computer has to take all this RGB information and translate it before it gets to the printer where it is printed in CMYK. This produces limitations in what colors can be accurately printed. For this reason, dark blues and bright reds look especially bad in printers. this graphic demonstrates the look of the color spectrum before and after printing. As you can see, the smooth color variations are lost.
This is why when you want to get an amazing print, it is important go to the professionals who know how to compensate for these color limitations. Modern professional printing processes can over come this and produce lifelike results if you have the knowhow and the willingness to invest in your image.