Lye + Water + Oils = Soap
Let's take a look at the ways these items interact with each other. By adding a solute (the lye) to a solvent (water), we create a solution. By contrast, the oils combine in a mixture. Finally, all of these come together in a chemical reaction to create soap.
In a solution, a solute is dissolved into a solvent. The key word here is dissolve. Typically a solute is a solid, although it can also be a liquid or a gas. But for simplicity's sake, let's assume it is a solid for our purposes, and that a solvent is a liquid. When these two items are combined, and optionally an outside force is applied to them (usually in the form of heat or stirring), they create a solution.
The big thing to remember here is that a solution does not chemically change either of the initial compounds. Think about a putting a teaspoon of salt into a cup of water. When you stir, the salt dissolves and disappears into the water. However, if you were to remove the water by evaporation or some other method, the salt would remain behind. Neither initial ingredient has chemically changed in any way.
In our case, both the lye and the water are polar substances, meaning that they will mix well together. Remember the principle of "like dissolves like," meaning that polar-polar and nonpolar-nonpolar solutions are possible. This is why we are able to create a lye water solution without a great deal of outside force. It is also why oils and water do not mix, since oils are nonpolar organic molecules. You would need some sort of nonpolar solvent, such as cyclohexane, to dissolve oils.
You can take a look at this comic by the Awkward Yeti to see this in action.
On the other hand, the oils we choose to combine for soap create a mixture. Similarly, a mixture does not change the chemical structure of it's ingredients. You may also separate them out of the mixture, although in our case it would be much more difficult than the example salt water solution.
It's closer to having a swimming pool filled with multi-colored confetti bits. You absolutely could separate out each individual color, but it would take a lot of time and effort to do so. Our mixed oils are similar to this confetti. We would probably need some specialized equipment and techniques to separate our oil mixture, but it could be done.
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By contrast, a chemical reaction creates an entirely new substance when it's ingredients are combined. Think of it like baking a cake. You combine flour, sugar, eggs, flavoring, butter, and other ingredients. Then with the application of heat from your oven, you have a new (and tasty!) substance. The chemical process of making soap is called saponification. This describes how the lye water solution reacts with the oil mixture to create soap. Just as in any chemical reaction, the initial elements must be carefully proportioned so that the final product will be usable. This is part of what makes soap making fun! However, unlike our cake analogy, we can't eat the soap....