When talking about processing in coffee we're referring to the steps that are taken between the coffee cherry being picked from the tree, the fruit removed from the seed, and the seed (or coffee bean) being readied for roasting.
The natural or dry process is regarded as being the oldest processing method, and in principal is the simplest. The coffee cherries are laid out to dry, often on patios in the sun or raised beds to allow more even air circulation. This drying process concentrates the sugars in the coffee, a bit like raisins or other dried fruit, and cause the coffee bean to absorb some of these inherent fruit qualities from the surrounding cherry. Once the coffee has dried it is then passed through a hulling machine to remove the fleshed polish the bean ready for export.
Natural process coffees are most common in dryer regions and rare in regions with higher humidity and rainfall where the coffee can more easily go bad from mould growth. It's most commonly associated with Ethiopia, Yemen, and certain regions of Brazil.
Because of its extended exposure to the fruit of the coffee cherry it can often have a distinctly fruity character, and can range from tasting like blueberries through to tasting like badly fermented wine.