Soaps are all salts of fatty acids of plant or animal origin and rarely contain additives beyond fragrances. Detergents are almost always made from petrochemicals and frequently commercial products contain a wide range of other chemicals such as preservatives, antimicrobials, surfactants and fragrances most of which are of questionable safety for internal use.
Soaps will typically contain stearate, palmitate (with a t), oleate or laurate, depending on the oil(s) they are made from. You will also see informal descriptions like tallowate, palmate, cocoate, palmkernelate, olivate, which specifically identify the animal or vegetable ingredients used (tallow, palm oil, coconut oil, palm kernel oil, olive oil etc.)
Hard soaps contain the above as sodium salts, liquid soaps as potassium salts.
Some soaps also contain glycerin, which is formed in the soapmaking process. This is removed from many commercial soaps, but is actually added to some ‘fancy’ moisturising toilet soaps. Handmade hobbyist soaps will contain the glycerin formed in the process.
I don’t btw have much sympathy for comments like ‘if you can’t pronounce it, don’t use it’. Most things we eat contain — naturally — substances whose chemical name would make you blink and which are essential, beneficial, or at worst harmless in the quantities that are consumed.