Saturation of color in photography shows vividness and purity of a color. In CMYK color mode, any color is a result of mixture of cyan, magenta yellow and black. Purity of a color is more, if black is less or absent in the mixture. So when black is reduced (in any photo editing software), the color becomes more saturated.
However, we cannot just reduce the percentage of black from any color. When black is reduced, the color moves towards "lightness". This lighter color is totally different in character from the original color. Therefore, the photo editing software compensate this loss of darkness by adding more of cyan or magenta or yellow (or mixture of two or all colors).
Depending on how the photo editing software is programmed, shift of saturation in color is slightly different form program to program. However, the basic programming of any software is done in such a way, that the tonal value of the original color shade remains the same. This means, that if original color is converted in black and white mode, its "grayness" remains the same, if we increase or decrease saturation of the original color, and then convert it in black and white color mode.
In photography, saturation of any color is subjective. When we are shooting a bright and live picture, we need siturated colors. On the other end, when the subject is calm and low key, we need less siturated colors.
Another consideration of situration of color is, whethere it is used in positive space or negative space. Positive space, or the main subject can be shot with more siturated colors, and negative space, or background should have less siturated colors. This will help to stand out the main subject against complicated background. However, the choice is subjective.
This example shows that the main subject, that is sofa set, is suppressed when the background has saturated colors.
Here the main subject stands out, when the background colors are less saturated.
Though saturation of any color is rectified during retouching and editing, we can control this saturation at the time of shoot. Point source of light gives more saturated colors in a photograph. Whereas soft light gives less saturated colors. However, this result is for general and overall saturation. For selective alteration, photo editing is a must.