Sodium hydroxide is also known as lye and caustic soda; it is highly caustic (able to burn or corrode organic tissue by chemical reaction) and reactive. Once the dry sodium hydroxide is mixed with a water a lye solution is created. This solution, when mixed with fats and oils cause a chemical reaction called saponification. The result of chemical reaction called saponification is soap. You can not make bars of soap without sodium hydroxide. If a bar of soap is really soap, sodium hydroxide must have been one of the ingredients that went into the process. Labeling laws do not require the listing of sodium hydroxide as a soap ingredient, so many soap makers will leave it off of their labels to not raise any questions with their customers. Soap by definition needs to have sodium hydroxide in it, whether it is listed on the label or not.
Most soap makers (including us) will superfat their soaps, which means there is an excessive amount of oils available to ensure all of the lye has been consumed in the chemical reaction called saponification. The inputs to the chemical reaction are oils and sodium hydroxide and the resulting products of the chemical reaction are soap and glycerin. This means there are extra free oils in the finished soap. This is good for two things. One, it assures all of the sodium hydroxide molecules can find an oil molecule to pair up with. It also leaves your skin moisturized after using it.