Black ink fades over time and it can be recolored using a similar shade of black ink. When black ink fades, it almost becomes blurry to the eyes and a tattoo artist can trace the existing outlines or fill in the black areas with new ink. Like any recoloring, you should only do so when absolutely necessary as the tattoo is harmful to the skin and will create scar tissue over time, so wait as long as you can to get the tattoo recolored with black ink.
White ink on a tattoo is very difficult to recolor. Over time, the white ink in a tattoo can become brown or take on your skin tone. When this occurs, you can attempt a recolor, but it will most likely not make much of an improvement. Keep the white part of your tattoo covered or apply liberal amounts of sunscreen to keep it white for a long time to avoid this issue. Since the white ink is not opaque like most tattoo ink colors, it is susceptible to this browning effect more so than any other color.
Retouching colored ink is common practice among tattoo artists. It is often best to go back to the original tattoo artist for the recolor, but if he is not available or you are not happy with the original piece, you can go to any reputable, licensed tattoo shop and have the color in your tattoo touched up. Like with black ink, only do so when absolutely necessary as the process of tattooing is harmful to your skin and if the tattoo is recolored too many times, scar tissue will form.
Another way to have a tattoo recolored is to have a cover-up performed on an old faded tattoo. In this process, you or a tattoo artist designs a new tattoo to go over the old one. You can use similar coloring so that the old tattoo cannot be seen behind the new design, or if it is faded enough, you can use all new colors for the new tattoo. Work your design well so that the old one is not visible after you have had the new design tattooed onto your body.