In simple terms, saponification is the name for a chemical reaction between an acid and a base to form a salt. When you make soap using the cold process soap making method, you mix an oil or fat (which is your acid) with Lye (which is your base) to form soap (which is a salt).
How exactly does this happen? In order to understand it, you must consider the chemical makeup of the acid and base being used in the reaction.
The base must always be composed of one hydroxide ion. For the most part, people use lye (one sodium ion and one hydroxide ion) as their base. You will notice that the sodium ion does not take part in the reaction at all. For this reason, other bases like potassium hydroxide can be used as well because it too is made up of one hydroxide ion. Potassium hydroxide is more prominently used for liquid soap making.
There are many different types of acids that will react with your base and saponify. Your acid could be olive oil, coconut oil or tallow among others. Each acid has a unique combination of triglycerides (compounds made of three fatty acids, attached to a single molecule of glycerol) which combines with the base (lye) differently. The amount of base needed to react with the acid will vary depending on the chemical makeup of the acid. See the saponification table help page for guidelines.
As you combine, and stir the carefully measured acid and base together, they start to react. The triglycerides within the acid release the single glycerol molecule (which turns into skin nourishing glycerin) allowing the fatty acids to combine with the hydroxide ions within the base, forming soap.
Two reactions occur. The first reaction is glycerol turning into beneficial glycerin, and the second reaction is the acid and the base combining to form a salt which is your soap.