It is an exothermic (gives off heat) chemical reaction that occurs when fats or oils (fatty acids) come into contact with lye (a base.) Saponification literally means "turning into soap" from the root word, "sapo", which is Latin for soap. The by-products of the saponification reaction are glycerin and soap.
Oils + lye (dispersed into water) = soap + glycerin
Oils and fats each have what is called a saponification value, which is the amount of lye needed to completely neutralize them into soap with no lye left over. Each oil has a different value, which is why its important to always run your recipes through a lye calculator.
The saponification generally takes about 24-48 hours to complete once the lye and oils have been mixed and the raw soap has been poured into the mold. This process can be sped up by adding more heat, or slowed down by keeping the process very cool.