It takes some kind of fat or oil to make soap. Beef fat is called tallow. Pig fat is lard. If you don’t want to use those, there’ s oil from coconuts, olives, soybeans. If you use olive oil, you’ve got Castille soap, because that’s what the Castillans used.
The greasy stuff is saponified by using an alkali such as lye, potassium hydroxide. The fats are broken down into glycerin and salts of the fatty acids, most popularly stearic acid. It’s the salts of fatty acids that do the cleaning trick.
If you add extra glycerin then you have glycerin soap. If, after saponification you add extra oils, then you have greasy girly soap. You’ll probably want to add lots of perfume to it too.
On the Beverly Hillbillies, Granny was a regular user of lye soap and it made her skin rough and leathery. “Thanks, it IS purty isn’t it?” she would say. Lye soap got this bad reputation because not all of the alkali got used up in the saponification process. The leftover lye was burning Granny’s skin!
One way to overcome the leftover lye problem is to age the soap for many months. This allows the chemical reactions to finish, and hopefully use up the remaining alkali. Once upon a time, the alkali was made by soaking wood ashes in water, and collecting the runoff. So the strength of the alkalinity was quite variable. Nowadays our chemicals are pretty consistent (there’s food grade lye for making hominy!) so recipes can tell you exactly how much to use. You won’t have any leftover chemicals floating about.
Until recently you could find lye in supermarkets and discount stores. Now you have to order it from accredited sources, and they’ll take down your personal info and store it for safe keeping. This is in case you’re a terrorist and are planning to wash things that shouldn’t be washed.
But I’ve never made soap, so I could be wrong about all of this. It’s just what I’ve been told.