Economically speaking, nations can be roughly classified based on the economic systems they have. Regardless of political preferences, prices in an economy may be set by the government (which would denote a planned economy) or perhaps by custom (which would denote a traditional economy). However, in a third system, prices may not be set at all, but may "float" based on the wills of supply and demand. This economic system, the market economy is that espoused (roughly) by America (but, these days, it's espoused - to some extent - by nearly all nations). Invisible Hand Theory, proposed by Adam Smith in the 18th century, helps to explain how the market economy that we all know and love works, even with its chaotic nature.
Mr. Smith reasoned that a market economy's resources were guided metaphorically by an "Invisible Hand." This wasn't the hand of God or some other deity, but simply represented pricing through competition. So, for example, a person might look at the competitive environment, see that he had an advantage in selling clothes, and therefore would enter some kind of industry related to clothes. Instead of the government (for example, a mandated quota) or custom (for example, a hereditary trade) dictating the best use of his resources, the individual would direct his resources based on competitive sense of supply and demand.
As with any economic system, Smith's Invisible Hand Theory and market economy attempts to answer three basic questions: what should be produced? For whom should things be produced? and how will things be produced? The Invisible Hand of competition proposed that these questions could best be answered by looking at the 'advantages' that different people and different resources have. The first kind of advantage that Smith formulated was the "absolute advantage" - simply, the person who can produce an item with the least cost has the absolute advantage, and therefore should produce in the industry and drive other less efficient people out of the market. So, if one person could make a pair of shoes for $3.00, and another person could make it for $.50 cents, then obviously, the person who could make the shoes for less would have a dramatic advantage, could make more shoes or reap greater profits.
According to Smith, nations or individuals would have few absolute advantages, and the market would always balance it. So, the shoemaker (who was good at making shoes) would make shoes for the benefit of the breadmaker (who was good at making bread, but not so good at shoes), and this would be more efficient than either making their own bread and shoes. If consumers didn't want shoes, for example, they would demand less shoes, and so the shoemaker might naturally have to redirect his efforts.
The first inaccuracy in his argument came with the fact that all nations are NOT equal - in our modern world, China and India have absolute advantages over any other nation in labor-intensive production, because they have the largest populations. However, it would not be economically sound for China and India simply to produce everything that requires manual labor. Imagine, for example, if the breadmaker was simply quicker and better at both baking bread *and* cobbling shoes. It wouldn't make sense for him to do all the work and the slower shoemaker to do nothing. So Adam Smith's absolute advantage theory was modified with the idea of comparative advantage, which compared two or more production rates. With this, resources were determined to be best spent based on comparisons: how many shoes the shoemaker could make *in terms of loaves of bread he didn't make* (or vice versa for the breadmaker: how many loaves of bread baked *in cost of the shoes he didn't make*. The benefit with comparative advantage was that, although one nation or individual could have absolute advantages in both bread production and shoe production, they would only have one comparative advantage.
The second inaccuracy in the invisible hand argument was that this hand would correct errors. However, without government restrictions, great abuses occurred unchecked. The free market, for example, couldn't say anything about worker safety rights, which were a cost to corporations (and thus a cut in profit margins). Additionally, the free market compounded failures (such as the Great Depression or even our recent housing crisis). So now, even in economic systems that are roughty market systems, the invisible hand is not left unchecked. Rather, the market economy is tempered with some government planning in order to curb excesses and deficiencies.