This is all about the way the coffee cherries are processed once they have been picked.
In a washed process the coffee cherry drives at the mill and is pulped to remove the toughest outside skin of the cherry and quite a lot of the fruit. Then the cherries are put in water and any that float are thrown away.
The cherries are then fermented (usually for 24-36 hours) either in water or on raised beds. Next, the remaining fruit of the cherry is washed off in large quantities of water. The beans are then dried to around 10% moisture (usually in the sun) and the remaining parchment (a creamy coloured layer left clinging to the bean) removed by a hulling machine.
Washed coffees often (but not always) have brighter flavours and lighter bodies.
In a natural process the coffee cherries are picked and then directly dried under the sun or on raised screens as full cherries. Once the cherry is dried, it turns into a dark brown pod that is hard to the touch; the green seed is taken out (by hand), leaving the other layers behind. This obviously has environmental advantages in terms of wage use but is more labour intensive. Sometimes a pulped natural process is used where the cherries are pulped without water or a machine is used to remove most of the cherry fruit before drying.
Natural coffees usually have more body and lower acidity than washed ones.
These processes are sometimes, not surprisingly, called wet and dry processing!