You might have heard about lye soap or soaps featuring various oils and butters like olive oil and Shea butter. All soap is made with lye (an alkali officially called sodium hydroxide). Without it, all you have is oil and perhaps some fragrance and colorants. The lye chemically reacts with oils (fatty acids) to give you the cleansing properties of lye without any of its harshness and the conditioning properties of the oils. This reaction is called saponification, and if you want the particulars of the chemistry, see Saponification in Wikipedia.
Depending on the oils used in the soap recipe, a soap can have a creamy lather, a lather with big bubbles, extra conditioning properties, or be extra hard. To encourage you to use handmade soap, compare the ingredients in a typical So and Soap with the ingredients in Dove soap. The only ingredients the two soaps have in common are water, sodium hydroxide (lye), and a natural whitener called titanium dioxide.