Stains are typically removed by one or more of the following strategies:

1. Dissolve the stain in a solvent. This is basically how dry cleaning works. A stain made of hydrocarbons (like bicycle grease) can be removed by a hydrocarbon solvent (like gasoline). Stains from fatty substances like butter and chocolate can be removed with organic solvents. The rule is "like dissolves like": pick a solvent that is similar to your stain, and you can wash the stain out.

2. Dissolve the stain using a surfactant. Surfactants allow water to wet fabrics better, and they can surround molecules in a stain and carry them into solution.

Soap is a surfactant; so are the sulfonates listed in the ingredients for many spot removers and carpet cleaners. A surfactant molecule contains long hydrocarbon tail with a small polar head. The hydrocarbon tail of the soap molecules surround (dissolve) grease, while the polar ends dissolve in water; the net result is that the grease/soap complex is water soluble and gets washed away. This process is called 'emulsification'. You can see it working if you add soap to some oil-and-vinegar salad dressing. The vinegar layer of the dressing gets cloudy because the soap has surrounded little droplets of oil and prevents them from rejoining the oil layer.

Sulfonates are more often used than soap these days. They work better in 'hard' water, which causes soap molecules to precipitate from solution (forming bathtub rings and 'soap scum').

3. Eat the stain. Oxidizing agents like chlorine bleach, peroxides, and borax attack the links that hold some long-chain organic molecules in stains together. The little fragments that are left are water-soluble, and they wash away.

Some cleaning agents contain enzymes that speed up reactions that digest proteins or fats in stains.

4. Hide the stain Some detergents contain 'whiteners' that whiten the fabric and do nothing to clean! They contain a material that absorbs ultraviolet light and re-emits it as visible light. Get a black light shine it on your clothes in a darkened room; if your laundry detergent contains whiteners, your clothes will glow!

Interesting question! Stain removers typically employ one or more of several mechanisms to remove stains. Stains are the result of an interaction between two chemically or physically dissimilar objects/materials. One common way to remove stains is by using detergents. Detergents usually have emulsifiers or surfactants that make it easier to remove the stain by coating the stain molecules and making it easier to lift them from the stained material. Another way to remove stains is by dissolving them in a chemically similar substance. Other methods include using enzymes to "digest" the stain. Another way yet is to simply "hide" the stain (e.g. bleach oxidizes the stain molecules so they appear lighter and less visible on a white shirt).