For such a commonplace chemical, soap is actually quite interesting in how it works.  It is actually made from fats, which are composed of a central triglyceride with three long chain, greasy fatty acids attached to it.  The fat is treated with a strong chemical base like sodium hydroxide.  This basic solution is called lye.  The lye breaks down the fats by hydrolyzing the fatty acids from the glycerol.  This reaction is called saponification.  The fatty acids then form chemical salts with the sodium hydroxide which makes crude soap. Perfumes and softeners are often added to make the finished product.

The action of soap is such that in water, the soap molecules form micelles.  These are spheres of conglomerated soap molecules where the greasy fatty acid chains are all in the interior of the sphere and the charged "head" portion of the soap is on the exterior.  So oils and dirts are trapped inside the micelles which freely disperse in the water.  The water with the dirty micelles is drained away, thus leaving behind a clean environment.

Solid soap is not really an emulsion or a solid salt.  It is best described as a cake.  When the soap cake is exposed to water, the charged surfaces of the micelles on the surface of the cake are attracted to the water and are freely solublized in it like salt or sugar would be.  So over time, more and more of the micelles off the soap cake are dispered in the water, thus wearing down the bar of soap until nothing is left of it.