At the core they’re all the same type of chemical, a long fatty molecule on one end with a charged up water-soluble tip.
This gives them the fancy ability to make oil dissolve in water by binding the fat end to oil and the charged end to water.
Then the different brands and uses add other things. Antibacterial soap adds alcohols to kill bacteria, pure soap just moves them around. Hand soap adds moisturizer so it doesn’t mummify your hands. Dish soap is heavy on the soaps and surfactants to make things dissolve in water that don’t want to dissolve in water.
Then a bunch of filler to make them watery or thick or solid or liquid or lavender scented or whatever.
Soap is a salt of a fatty acid.
A fatty acid is (in ELI5 terms) a molecule which has a bit of oxygen on one end attached to a loooong chain of carbon and hydrogen.
A salt is a compound where a negatively-charged molecule and a positively-charged molecule link up together. In this case, the negative charge is on the fatty acid; the positive molecule is generally sodium or potassium in household soaps, but can be calcium, lithium, aluminum, or many others.
Because soap is defined by these long-tail molecules which are special on one end, they have special properties. The tail likes to attach to oils; the other end likes to attach to water. This makes it cluster around droplets of oil and other dirty stuff and isolates them, which makes them easy to wash away.
Beyond that, you can add other materials to change a soap’s properties and thus its type. Using different oils can make your soap liquid rather than solid, a special dispenser makes it foam up, you can introduce air bubbles into your bar to make it have different physical properties, you can add a disinfectant to get antibacterial soap.
What’s common to all soaps is they are long molecules with one end that attracts water and another end that attracts oil.
So you put soap on something dirty and the soap particles stick to oil/fats/grease, then when you rinse with water the soap particles get pulled with the water and pull the oil/fats/grease off with it.
Cell membranes are also made of fat, and the soap molecules sticking to them break them. So really all soap is antibacterial to a degree.
Most of what we call “soap” would actually be a detergent in the chemistry sense. Dish “soap”, laundry “soap” are actually detergents. Soap, as others specifically have said, is usually some sort of fat treated with a strong base like lye to create the bars we can use for cleaning. The specifics of what detergents and soaps do, in the most basic sense, is to help break the bonds between water molecules so other molecules, like grease and dirt, can fit between them and wash away.
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