Surfactants are a class of molecules composed of a hydrophilic "head" (water-loving) and a hydrophobic "tail" (water-fearing). This particular structure allows them to have many wonderful properties, that I will maybe explore in the future if someone is interested but that I will spare you at the moment otherwise bye-bye quick read.
Now I would like to focus on just the fact that these molecules have hydrophobic tails: when dispersed in water, the tails will try to stay as far away from it as possible: they will shield each other and pack onto other hydrophobic molecules, such as the fat of our skin or our dishes. The tails will arrange all around this fat, exposing their hydrophilic heads to the water.
The net result of this mechanism is twofold: on the one hand, the grease molecules will be covered by the hydrophilic heads of the surfactants, thus increasing their solubility in water; on the other hand, since the hydrophobic interaction is very strong, the "will" of the surfactant tails to cover the grease will cause it to detach from the surfaces, facilitating its removal. We are, in simple terms, cleaning.
What is the difference between "natural" and synthetic surfactants?
Refraining at this time from any invective against the use and abuse of the term "natural", usually are called "natural surfactants" those surfactants that are obtained from raw materials of plant origin through "simple" processing: we can range from minimally processed surfactants, directly extracted from plant species naturally rich in saponins, all the way to surfactants that are obtained by reacting compounds from plant species. The most famous representatives of this last group are Decyl Glucoside (also known by the splendid marketing-friendly name of Coco-glucoside*) and Lauryl Glucoside, both obtained from the reaction of sugars from corn and fatty alcohols from palm oils.
These surfactants have the characteristic of having a relatively low cleaning power. They are particularly suitable in fields such as cosmetics, where an excessive cleaning power could lead to irritation.
Synthetic surfactants, on the other hand, are... well, everything else.
They are a vast class that includes very different molecules, obtained from different sources through different processes. It is therefore extremely difficult to give one-fit-all description. Let's say that, when we refer to them in the cosmetic-detergents field, we usually mean molecules where the chemical processing has been greater, perhaps leaving the dictates of Green Chemistry. This class of surfactants usually has much higher cleaning and foaming performance, as well as very low production costs: for this reason it is widely used in cosmetics, household detergents and in a host of other fields. Typical examples of this class are SLES (Sodium Lauryl Ether Sulfate) and its little brother, SLS (Sodium Lauryl Sulfate).
FUN FACT! Despite being in recent years pointed at as the devil, both SLES and SLS are actually derived from.... drum roll.... coconut or palm oil, just like their "natural" cousins. The difference is that, to obtain SLES, a reaction called ethoxylation has been performed. This reaction involves the use of ethylene, a compound that to date is produced mainly from natural gas and hydrocarbons. However, it can also be produced by dehydration of ethanol from vegetable sources, making even the infamous SLES absolutely... "green".
(*): Decyl Glucoside (60% C8-C10,40% C12-C14); Coco Glucoside (40% C8-C10, 60% C12-C14)