Soap occurs as a byproduct of a reaction between fats and oils and lye in the presence of a small amount of water.

To make the best soap we must begin with the selection of these fats and oils with a few key considerations. Not every fat is created equal or that is to say that some fats will impart a much different characteristic on your soap than others. These depend on the molecular composition of those fats and the physical properties imparted by those molecules but more on that later.

First we must prepare our equipment. We are going to need a few basic things for a Hot processed soap.

Step 1: Equipment

Some soap makers use equipment that is devoted only to the act of soap making. This is of course your decision but the potential contamination of lye is dangerous. What you need will be...

A hot plate or a heat source that can heat your fats and maintain them at a an even temperature of 45 to 54 degrees centigrade.
A vessel to heat your fat in as well as to mix your ingredients. We will be using a one liter beaker since this is SCIENCE!
A scale to weigh your fats.
A thermometer.
A method for mixing your soap. We are going to use a stir bar since our hot plate has this function. This is essentially a magnetically driven spinning bar at the bottom of the beaker.
Molds for your soap. When you are finished with your soap you will need to place it in a mold to finish curing until it is ready to use. This could be any shape but the soap needs to come back out once you've placed it there.

Step 2: Your Ingredients.

Fats and oils - The use of a soap calculator that is available online will help you with the selection of your fats and oils. A good soap is going to be comprised of a mix of a variety of fats and oils depending on the quality of the soap you would like.

I will explain more about the soap calculator but for now here is a link to the Soap Calculator
I chose to use Crisco vegetable shortening and palm oil for my soap.
The soap calculator includes suggested values for the recipe and the individual values for each fat in the following categories; Hardness Cleansing Condition Bubbly Creamy Iodine INS.
We are not going to be concerned with the Iodine number but this is essentially a measure of the amount of unsaturated fatty acids in a fat. The higher the iodine number the softer the soap.
Each fat has a unique profile and the soap calculator will also calculate the values for your entire recipe when you adjust the proportions and calculate your recipe.
Adjust your recipe until you find the qualities you would like. My total score for 50% Crisco and 50% Palm Oil is 130 INS, less than the ideal 160 INS but it will work for now.

Step 3: Lye or Sodium Hydroxide

Danger! Lye is a very caustic substance and will burn you if not handled properly. Keep some acetic acid or vinegar handy in case of a spill so that this strong base can be neutralized. Lye is also used as a drain opener because of its ability to eat through organic material. This could be your hand or eyeball!

Lye can be purchased at your local hardware store or ordered online through one of many soap making supply companies.
However you obtain the lye it will need to be mixed with the water before adding it to your fats.
The lye must be added to the water and never add water to the lye.
You will use about 38 % of the weight of your fat for your water weight. The amount of lye you use depends on the saponification value of your recipe as provided by the soap calculator.

Step 4: Fatty Acids: Some Science

Fatty Acids:

The qualities of the soap you make will depend largely on the properties of the fats. This in turn depends on amount of saturated and unsaturated fatty acids in your fat.
A fat is said to be saturated when the hydrocarbon chain consists no double carbon bonds. That is to say that every carbon is sharing a covalent bond with four other molecules and there is no place for more hydrogen atoms in the hydrocabon chain.
The unsaturated fatty acids have less intermolecular attraction than saturated fatty acids because their shape is irregular. Olive oil is a great example of an unsaturated fat. The unsaturated fatty acids and their weak attractions cause olive oil to be liquid at room temperature and to contribute to a softer soap.
Saturated fats have strong intermolecular foces because of their more regular shape. This causes fats like butter or crisco to be solid at room temperature and contribute to hardness in a soap.
Pick your fats wisely!

Step 5: Saponification Reaction and SAP Value

Saponification is the process where lye essentially rips apart the triglycerides that make up fats. A triglyceride consists of three fatty acids and a glycerol molecule assembled through a condensation reaction in a plant or animal. The hydroxide acts via hydrolosis to remove the fatty acids from the glycerol molecule leaving a water molecule behind for each fatty acid removed.

Sodium Hydroxide in solution breaks apart the triglyceride
The glycerol molecule is liberated and floats off into solution
The Sodium ions bind to the carboxyl ends (COOH) of the fatty acids to make soap molecules.
When this reaction has been completed your soap is finished.

The hot process (application of heat) expedites this reaction but it will still take a few days before your soap will be ready.

SAP VALUE - This number as provided by the soap calculator is the number of grams of lye needed for each gram of your fats and oils. This is a ratio. For my recipe the SAP value is 0.137 which means for every one gram of my fats I will need 0.137 grams of NaOH or sodium hydroxide to create my soap molecules.

Any excess sodium hydroxide or lye in your recipe will create a soap that has the potential to irritate your skin or even cause a chemical burn. Do not add to much. Do your math.