Why is color management important?
Color management is the process of capturing a color using an input device, displaying it on a screen, and printing it all whilst managing the exact colors and the range of colors on each medium.
The need for color management is probably explained best with a photograph of a bird on a frosty day in winter.
Displays typically over-saturate the blue channel, making the images look cold.
Notice how the white is not 'paper white' and the black of the eye is now a muddy brown.
The basic problem we have here is that each device is capable of handling a different range of colors. So while you might be able to take a photo of electric blue, most printers are not going to be able to reproduce it.
Most image devices capture in RGB (Red, Green, Blue) and have to convert to CMYK (Cyan, Magenta, Yellow, and Black) to print. Another problem is that you can't have white ink, and so the whiteness can only be as good as the paper color.
Another problem is units. Without specifying the scale on which a color is measured, we don't know if 100% red is near infrared or just the deepest red ink in the printer. What is 50% red on one display is probably something like 62% on another display. It's like telling a person that you've just driven 7 units of distance, without the unit you don't know if that's 7 kilometers or 7 meters.
In color, we refer to the units as gamut. Gamut is essentially the range of colors that can be reproduced. A device like a DSLR camera might have a very large gamut, being able to capture all the colors in a sunset, but a projector has a very small gamut and all the colors are going to look "washed out".
In some cases we can correct the device output by altering the data we send to it, but in other cases where that's not possible (you can't print electric blue) we need to show the user what the result is going to look like.
For photographs it makes sense to use the full tonal range of a color device, to be able to make smooth changes in color. For other graphics, you might want to match the color exactly, which is important if you're trying to print a custom mug with the Red Hat logo that has to be the exact Red Hat Red.