How does a facial cleanser work on the skin?

The point of a cleanser is to clean off the oil, debris, and makeup that accrue during a normal day, and to do that you don't need anything harsh or too powerful. All soaps or detergents work by a process called "micellation," which breaks down big substances (such as oil or dirt) into smaller ones using a chemical reaction, so they can rinse off the skin easily. A fatty-acid ingredient attaches to the oils on the skin and dissolves them, while an alkaline element makes it possible to rinse it all off the skin. Micellation works best using warm water, about the same temperature as your body, simply because it's more compatible with the cleanser and with your skin.
Traditionally, soap is made from either vegetable oil or animal fat and lye (otherwise known as sodium hydroxide, a water-soluble alkali that emulsifies in water). Today, most cleansers (including products that are loosely defined as "soap") use synthetic detergents, which are cheaper and easier to manufacture and tend to be gentler because they are less alkaline. The normal pH of the skin is between 4.5 and 6.5, and an alkaline substance raises that level, increasing the potential for dryness and irritation. They both work in the same way, which is why many cleansers leave the skin feeling dry, since their job is to literally wash away oils from the surface.