To make soap, we must break the fat into its fatty acid and glycerol constituents. The fatty acid has a long hydrocarbon tail which is soluble in fats, and a polar oxygen end which is soluble in water. Thus a fatty acid in solution acts as a soap by dissolving fats in one end of the molecule and water in the other. When we use a strong base, such as lye to break apart or hydrolyse the fat, the fatty acid is present as a large cation which is polar at one end and non-polar at the other. Just as we can have sodium chloride and sodium carbonate which are soluble in water, we can have sodium octanoate, the sodium salt of octanoic acid, which is also soluble in water.
Saponification is the term applied to the hydrolysis of fats using a strong alkali like lye.
While this reaction may appear intimidating because of the long formulas, it is, in fact, quite simple. This is simply specifying the particular fatty acids present in the soap. With enough soap, these fat globules become covered with a water-soluble coating and disperse throughout the solution, as in the last figure. They are not truly dissolved since individual fat molecules are not dispersed in the solution. Rather, we say the fat is emulsified .