What does "natural" actually mean?

Short answer? Not a damn thing. The term is pretty much meaningless in the US; at best it only means that the product doesn't have added colors, artificial flavors or synthetic "stuff". From the FDA:

What is the meaning of 'natural' on the label of food?

From a food science perspective, it is difficult to define a food product that is 'natural' because the food has probably been processed and is no longer the product of the earth. That said, FDA has not developed a definition for use of the term natural or its derivatives. However, the agency has not objected to the use of the term if the food does not contain added color, artificial flavors, or synthetic substances.

To expand on Jolene's answer, there is not only no official definition, but the only definition which fits its common usage is A food which a certain group of persons is not afraid to eat.

Philosophically, "natural" is the opposite of "artificial" or "man-made", but philosophy doesn't give us a limit of interaction under which something stays "natural. Does a ear of corn stop being natural when you pick it? When you remove its leaves and silk? When you mill it into cornflour? When you make HFCS out of the cornflour? Somewhere along the process, it becomes "unnatural" as the term is commonly used, but there is no technically obvious place for placing the turning point.

There are three possible ways to define "natural" which would make it more or less objective, but they don't cover the word as it is used in real life, even though there is overlap. Besides, if one of them were the "correct" definition, the commonly assumed connection to "healthy" or "better quality" would not automatically follow from any of them.

• The first candidate definition is to say that non-processed food is "natural" and processed food is not. But there are many counterexamples to it. For example, I've seen many foods listing "natural fruit sugar". Well, the fructose in them is processed to about the same level as the sucrose in the competing products, but I've never seen somebody include white refined sugar in their mental list of "natural sweeteners".

• Another candidate would be "synthetic", as in chemically synthesized by man as opposed to extracted from an organism which produced the molecule. It would have the advantage of being consistent with the use of "natural" in textiles, another major area of everyday life. Again, this is not congruent with real world usage - a sizable proportion of food additives, which in my experience scare the average natural-eater, are extracted from plants and bacteria, for example xanthan or MSG.

• The third possible definition (added after Steve Jessop's comment) would look at human history, draw a limit somewhere and say "these foods are natural, the others are a product of civilization and thus unnatural". As pointed out in the comments, this is indeed used in some nutrition theories such as Paleo. But when we ask ourselves where to draw the line, we'll notice that the average person talking about "natural" food is not as radical as the Paleo people, and will see a loaf of whole wheat bread as a very natural food. Could we find a point in more recent history which supports such a division? It's hard, as technology has evolved in a continuous way, but the best candidate would be the Industrial revolution, amounting to a rapid switch from low-tech to high-tech. Still, if we apply this definition, we see that foods like smoothies or baking soda would fall on the "unnatural" side. But in common use, baking soda has an old-fashioned feel and, if we look at cleaning methods, is often touted as a "natural" alternative to purpose-created cleaning products. So, this definition again doesn't explain the common observations.

"Natural" is an ideological construct common in our society. As with other ideological constructs, its true meaning is determined by who is saying the word, and what his attitude to the object of speech is. As far as I have observed it, it is not connected by any physical properties of the object, at least not in a consistent manner. This is also why there is no definition by the FDA, and there cannot be one either, at least not one which covers the current use.

Note that despite the condescending tone above, I also prefer eating an apple over eating a mix of fructose, water and processed fibre, just like the people who insist on "natural" food. I fully appreciate that having a term which helps us decide the relative "healthiness" of a food would be useful for society. It's only that "natural" isn't that term, even though people insist on using it that way.

Narrowing the question to just Natural Flavor, we have Code of Federal Regulations Title 21:

(3) The term natural flavor or natural flavoring means the essential oil, oleoresin, essence or extractive, protein hydrolysate, distillate, or any product of roasting, heating or enzymolysis, which contains the flavoring constituents derived from a spice, fruit or fruit juice, vegetable or vegetable juice, edible yeast, herb, bark, bud, root, leaf or similar plant material, meat, seafood, poultry, eggs, dairy products, or fermentation products thereof, whose significant function in food is flavoring rather than nutritional. Natural flavors, include the natural essence or extractives obtained from plants listed in subpart A of part 582 of this chapter, and the substances listed in 172.510 of this chapter.

Modern fermentation products include a multitude of sins.

It gets even more technical:

(iii) If the food contains both a characterizing flavor from the product whose flavor is simulated and other natural flavor which simulates, resembles or reinforces the characterizing flavor, the food shall be labeled in accordance with the introductory text and paragraph (i)(1)(i) of this section and the name of the food shall be immediately followed by the words with other natural flavor in letters not less than one-half the height of the letters used in the name of the characterizing flavor.

Long ago I learned what the Natural Flavorings in Mapleine were, but can no longer remember, other than that they had not the slightest to do with Maple trees.

By a strange coincidence, NRC (a Dutch quality newspaper) published a report this weekend (7 Feb 2015) that EU law allows flavors created by genetically modified yeasts in bio-reactors to be described as "natural" (since yeast fermentation is a a natural process). Link. Unfortunately the article is in Dutch and for subscribers only, but the title and header are (translation by GlobeFish):

Grapefruit flavor from the bioreactor

'Natural' is what consumers want. But the meaning of that word is stretched. 'Natural' flavors and fragrances now come from the lab.

One highlight which appealed to me: "orange drink with natural flavor" could mean bio-reactor, but "natural orange flavor" means that real oranges were used.

From Consumer Reports:

Most consumers would probably be surprised to learn that the FDA has not developed a formal definition for use of the term "natural" or its derivatives. But the agency has not objected to the use of the term if "nothing artificial or synthetic (including all color additives regardless of source) has been included in, or has been added to, a food that would not normally be expected to be in the food" — though these are still found extensively in "natural" labeled foods.

The USDA, which regulates meat and poultry, says that a product is natural if it contains: "No artificial ingredient or added color and is only minimally processed. Minimal processing means that the product was processed in a manner that does not fundamentally alter the product."

The problem is that if a small tribe crushes and sun dries a beetle with a stone or heavy wood and ads it to a dish for colour then it would be 'artesenal' where as if the exact same outcome was achieved in a machine production line it would be called 'processed'.