Value is essentially the darkness of the pigment. Less value equates to a darker color. If you take a color and remove all hue, you are left with value - basically greyscale. Brightness is another term used for value. Often brightness is a bit easier to remember since more value means a "brighter" color.
Basic Value scale
Saturation is essentially the depth of the pigment. More saturation means more pigment. Less saturation equates to a tint or lighter shade of the color. Luminosity is another term used for Saturation.
Basic saturation scale. (based on a red hue)
Hue is the base pigment.
Basic Hue scale
"Color" would actually be the combination of all three of the above.
For example if you have a Hue which is red.
- Decreasing Saturation/Luminosity will cause the red to start moving into the pink areas. Hue minus Saturation = lighter color
- Decreasing Value/Brightness will cause the red to move into the maroon or burgundy range. Hue minus Value = darker color
Anther way to think of the relationship is to consider removing Saturation as adding white to the hue. And removing Value as adding black to the hue.
If you are coloring a greyscale image, you would use the greyscale to designate the value of the colors. To that end you would never adjust the value of the color you are using allowing the underlying greyscale tones to create darkness. You would choose a hue and a saturation, then paint over an greyscale area, setting the blending mode to Multiply or Darken to allow what you are painting to interact with the underlying image. Because you are using a base greyscale image, you would never want to create an area which is lighter than the image. The only variation would be how much color is applied, thus hue for the pigment value and saturation for the amount of pigment.