Ever hear, "Yeah, that shop has a really nice natural right now, but I'm more of a washed-coffee person," and feel totally out of the loop? Well heyooo, that's what I'm here for.
First, it's helpful to know that coffee is a fruit. It looks like a cherry, it grows on trees that can get pretty tall. What we know as "beans" are actually basically cherry seeds with a kick.
To get to these seeds so that you can roast, brew, and drink them, you can go a couple of ways. For most of coffee's history, it was processed "naturally," meaning ripe cherries were harvested, then the fruit was laid out to dry. Actually, the first human consumption of coffee wasn't in the form we know it today--East African tribes made energy bars out of dried coffee fruit and animal fat (yummy!). People still consume coffee fruit without the seeds, in granola mixes, or as tea.
In more recent years, thanks to advances in agricultural technology, we've started producing "washed" coffees. This means that instead of laying the fruit out to dry right after harvest, the cherries are instead sent through a crusher (also called a "pulper" or a "mill"), to get the seeds out.
Now, you know how when you eat a cherry, the seed is all slimy and still has some fruit on it? Same with coffee. To get that stuff off, the newly-skinless beans go into fermentation tanks, which look like big concrete square pools. Then it basically rots for a few days.
Letting coffee sit and ferment like this helps break down the slimy layer, which is washed off in a series of water rinses. After 2 to 3 washes, most of the fruit stuff is off, and coffee can be laid out to dry. So the basic difference between "natural" and "washed" is dried in cherry vs. cherry removed before drying.
So why does this even matter? It depends on what you like! Washed coffees are not anymore hygienic or healthful than natural coffees--they're simply processed this way because people like the clean, bright taste of washed coffees. Some also use wet processing because natural coffee looooves to rot and the flavors can be earthy and unpredictable.
There you go, now you know!