Colors can be removed chemically or physically.  In the first case, the dye molecules remain in place on the shell, but are altered enough that they no longer reflect light.  In the second case, the molecules are actually removed from the shell of the egg.

There's a half dozen different ways to remove color; rinses remove color to allow otherwise incompatible dyes to be used in a series, while soaps/detergents, acid etching and bleaching are used to return the shell color to white.

DYE RINSES:

Each successive dye color DOES remove its predecessor, to some extent. Obviously, yellow doesn't dye well over black. The traditional color order goes from light to dark to take advantage of this feature. Some dyes, though, can be utilized to remove colors and switch between color families.

Orange Rinse: Orange dye does a particularly good job of removing color, at least in part due to a lack of vinegar (UGS Orange); many pysankarky keep a separate batch of "orange rinse" just for that purpose. It is used to go form the blue/green family of dyes to red/orange.

Yellow Rinse: Yellow is not as fast as orange at removing dyes, but will do so.  It is useful for moving from the Red/Orange family of dyes to the Blue Green.

Gold Rinse: UGS Gold (without vinegar) can also be used to switch from the Red/Orange family of dyes to the Blue Green.

Pink Rinse: UGS Pink is said to do a great job of covering green.

WATER RINSE:

Dye molecules remain in equilibrium between those adherent to the eggshell surface and those in dye solution.  Placing a dyed egg into water will dissolve the bonds to the shell and put the molecules back into solution. This is a slow process.

A rinse under plain water will remove most dye, although actually soaking the dyed egg in a cup of water is more effective. The shell usually won't go down to a pure white, but may remove enough dye to allow you switch between color families in many cases. Use tepid, not hot, water!

SOAPS/DETERGENTS:

Soaps and detergents will bind with various dyes and remove them from the shell. After using ANY soap or detergent on an egg, the egg should be rinsed well with plain water and dipped briefly into vinegar to reset the pH before any further dyeing.

Dish Soaps: Washing a dyed egg with Ivory liquid and water will do a better job than rinsing with water alone. Dawn dish washing liquid is also good at removing pysanka dye.

All Purpose Cleaners:  Rinsing with Simple Green cleaner (or Mrs. Meyer’s all-purpose cleaner) works better than dish soap. Spray on full strength and rub it around with your hands. Scrubbing is not necessary, but you might want to use a soft toothbrush to get into all the nooks and crannies. Again, rinse well and dip briefly into vinegar before dyeing the next color.

Some people have great results with this process, but I have difficulty getting good dyeing after using soaps or detergents.  In my experience, later colors tend to be pallid or pastel.

ACID ETCHING:

Acids are used to remove the dye and a thin layer of shell itself. This is the traditional method of creating "white" pysanky. My ancestors used mild culinary acids for this purpose: vinegar, sauerkraut juice. The pysanka would be soaked in such a solution for hours, until the surface was slightly etched and the dye removed.

The traditional way is quite slow. Using concentrated acids is much quicker. I use Acid Magic, a fumeless version of muriatic acid (sold as a pool cleaner); others use CLR, toilet bowl cleaners and other commercially available acids. I apply to the shell with a "puffy" brush, let it fizz a bit, and rinse. And repeat several times. A soft brush should be used to help remove the debris when rinsing.

This method should only be used on empty egg shells, not on full eggs. It should be noted that colors do not take as well on etched eggshell as they do on an intact one; etched eggs will often dye poorly (goose eggs perhaps excepted).

BLEACHING:

Bleach does not actually remove dyes; it breaks chemical bonds within the dye, changing the molecule so it no longer reflects light.  In essence, the dyes become invisible.

Bleaching is best used to remove dyes when you want the final background color to be white. It disrupts the protein molecules on the egg’s surface, and makes the shell difficult to dye afterwards. The eggs should be washed well after bleaching, to get rid of the sliminess of the bleach (and all remnants of it). It should be allowed to sit and dry well before anything more is done (e.g. wax removal).

Bleached eggs do have a tendency to yellow over time (as do fabrics that are bleached).