At the beginning this process was done by cutting up different materials in the shapes needed for a particular color on the image. Once that was done, plates were made with different ridges or empty spaces. Those empty spaces would then be filled with the appropriate color and the image would be printed in full color.
As time went by, people began to use other mediums such as an airbrush, or watercolors. These would then be photographed and printed. Doing this allowed for more subtleties in the end product, since you could now do things like blending colors to give it a more realistic look.
With the type of technology available to us now, this process is now done digitally using layering or editing programs such as Photoshop. With these programs, colorists could build up layers of all the different colors or textures and edit each little part easier.
I have made a short tutorial explaining how I went on to apply this process to a black and white drawing using Photoshop here.