The Marriam - Webster dictionary defines soap as: a cleansing and emulsifying agent made usually by action of alkali on fat or fatty acids and consisting essentially of sodium or potassium salts of such acids
A simple definition is soap is simply a substance made of oils or fats mixed with a strong alkali. Like sodium hydroxide or pot ash.
That being said, ALL hand made soap is made with lye - even melt and pour soap (usually known as Glycerin soap). Technically Cold Processed Opaque soap is glycerin soap but that's a conversation for another time.
So the process of turning oil and lye into soap is called SAPONIFICATION.
What is Saponification?
The web definition: a chemical reaction in which an ester is heated with an alkali (especially the alkaline hydrolysis of a fat or oil to make soap)
This happens in stages. Just too uninteresting to spell out here and time consuming to break down.
So in my words: when you mix the oil with the lye solution it heats up and begins to thicken. This process is Saponification. The soap can heat up to 200 degrees and will (Gel) turn translucent before it eventually cools and settles on being opaque.
Is this important to know? I think it is. I formulate my soaps by the fatty acid properties the oil has. I do believe knowing the chemistry of soap making will ultimately help you make a better bar of soap. While I didn't outline the specifics of the chemistry of soap making, it is something worthwhile to look into.