There are two main methods to get the coffee out of the cherry - washing the cherry or not washing it - and there are many variations in between. In the washed process first you remove the fruit by putting it through what looks like a large grater to remove the fruit and then you wash it in a large tank.
A fully washed coffee will accentuate a brighter, cleaner and lighter taste of coffee.
A natural coffee will not be washed at all but have the whole cherry (with the coffee inside) dried and then the dried fruit is removed later from the green bean. The natural method will accentuate body, minimize acidity and tend to have more of a fruity taste but there may be more defects because the coffee isn't washed. This is because defective beans float in water and there may be issues with rotting cherry fruit if it's not tended to properly.
A pulped natural is between a washed and a natural. In this process the producer will remove the skin off the cherry but leave the fruity mucilage intact during drying. These coffees have more body and lower acidity than washed but are cleaner and more uniform than natural coffees.
Then we have a wet-hulled sumatra which is essentially a different take on the pulped natural method in that they remove the green bean from the fruit in a similar way as is done in the natural method but then instead of drying it they put it in a tank or sack and let it ferment overnight and then the rest of the cherry is removed. This results in a funky musky, spicy, full body and low acidity coffee which Sumatra is famous for.
left: washed Colombia, center: pulped natural Brazil, right: wet-hulled Sumatra.