How Stain Works
Wood stain is composed of pigment, solvent and binder. The solvent helps the pigment and binder sink into the pores of the wood; the binder attaches it to the walls of the pores. When the solvent evaporates, you're left with permanently stained wood. To get rid of the stain in the wood's pores, you'll have to remove some of the wood itself. Before setting out to remove stain, decide whether it's worth it. Most old cabinets can be easily refurbished by wood cleaners that remove the built-up wax and grease or can be painted to cover the old finish. If you just have to get rid of all of the old finish, though, plan your strategy before starting.
Strategize Before Starting
Take all the cabinet doors down and remove hardware. If you'll reuse the hardware, give it a good soak in hot, soapy water and scrub it with an old toothbrush. Examine the wood in the cabinet doors. If they're veneers (thin coatings of hardwoods over a softer wood base), you may want to reconsider stripping and just clean using one of the low-moisture wood cleaners recommended for veneers. Check to make sure that the wood is sturdy and replace or secure loose or missing sections.
If you still want to remove the stain, you'll have to strip using a product recommended for indoor use and then sand the surface to remove the rest of the finish. Remove as much finish as possible, then apply a second layer of stripper, cover it with waxed paper, and let it sit for a while before wiping or rinsing. The reintroduction of solvent may pull up some of the binder and pigment near the surface. Sanding with an orbital sander using medium-grit paper should remove most of the stain.
Alternative treatments include wiping with mineral spirits (a solvent) or bleach or ammonia (caustics) to blanch the surface of the wood.
Exterior "stain strippers" are formulated to remove exterior deck finishes, but before trying any of these, get some advice from someone who's used them. They can only be used outdoors---most of them require a neutralizer and a thorough flush with water.
Finish the Job
Chemical stripping and sanding are hard on wood surfaces; once the stripper is removed or sanding is complete, the wood surface will need to be repaired before a new finish can be applied. Sand the surface with progressively finer sandpaper and apply a wood sealer to protect the wood. The surface is now ready---minus a layer---to refinish. Even with heavy sanding, most wood will retain some of the stain's color. Choose a stain that works with the original color and test it on an inconspicuous part of the cabinet before staining. Often, wood in the cabinet body is different from the doors; be prepared to correct tints to match doors to bodies.