In August of 2011, the People's Republic of China launched an ex-Soviet aircraft carrier on a test voyage, marking China's entry into the group of nations that can operate an aircraft carrier. This was widely seen as a milestone, but the full significance of China's accomplishment can only be viewed in the context of China's past and future, as well as some technical information on what exactly an aircraft carrier is.
First off, it is somewhat questionable whether China could be said to have an aircraft carrier. After all, an aircraft carrier carries aircraft. The ex-Soviet carrier, called the Varyag, was designed and half-built for that purpose, but was sold to China in 1998 half-completed, and was never an operational aircraft carrier. The Chinese have made it seaworthy, but have yet to use it for its intended purpose.
It should also be noted that the term "aircraft carrier" can refer to many vessels that are in many ways very dissimilar. The United States and (to a much lesser extent) France are currently the only countries that have aircraft carriers big enough to handle general purpose aircraft. All other countries, including Russia, from whom the Varyag was purchased, operate much smaller aircraft carriers that carry V/STOL aircraft that are designed to work with smaller decks, but that have sharply curtailed abilities because of it. So even when (and if) the Varyag becomes operational, or the Chinese build more aircraft carriers based on its technology (which they are planning to do), they will still be smaller than American aircraft carriers, and designed for specific naval missions, rather than being bases for striking ground targets or establishing air superiority. And this is assuming that not only the aircraft carriers are built, but that the personnel necessary to operate the naval and air components of the carrier are also trained.
That being said, a bit about China's military situation should also be explained. In the past two generations, China has fought wars with India, Vietnam and The Soviet Union, although none of these "wars" were really a matter of existential threat. They are also matters that for the most part are not dependent on naval power (although I would find it impressive if China could send an aircraft carrier into the Himalayas to settle its border dispute with India). One of China's largest foreign policy issues is its claim of ownership of the island of Taiwan, although China's refusal to act on its claim is probably due more to diplomatic constraints than military ones. Japan and The United States are also long term strategic threats, but again, not currently existential ones.
So the Chinese don't really need an aircraft carrier, and they can barely build or operate one. What then is the reason of the Chinese aircraft carrier program, evidenced by the refitting of the Varyag and the stated intention of building several additional carriers? It seems to be slightly irrational, and that is indeed the point.
Post-World War II, probably the foremost reason that any nation (including the United States, whose carriers are actually extremely capable of carrying out military operations) has deployed aircraft carriers has been the prestige of owning and operating an aircraft carrier. They are such expensive, technically complicated and improbable pieces of engineering that being able to operate one says much for a nation's economy and technical prowess.
And if that was the purpose, it seems to have been fulfilled. Although most experts took a reasoned view to the Chinese carrier program, there was alarm from some quarters. Although it is perhaps a strawman to use it as typical, one of my favorite articles on the matter came from Snopes, which showed a reputed Chinese carrier in development that was a gigantic, twin-hulled design with twin runways, able to handle at least three dozen "stealth fighter-bombers" and AWACS aircraft, and just to top it off, capable of twice the speeds of any United States carrier. This somewhat exaggerates the technology actually available, unless the technology is Photoshop.
Naval architecture is hardly my area of expertise, although my actual level of expertise, education policy is very relevant to this discussion. I have noticed that many Americans have come to think of the Chinese educational system as far superior to our own, and to think of Chinese children as elite calculating machines, able to handle calculus in middle school. While this is probably a better stereotype to have then the stereotype of past generations, when all Chinese were superstitious farmers, it is still an inaccurate stereotype. China's educational system is producing some great results in the urban centers, both in terms of a widely educated populace and engineering feats, China still has a large rural population with a poor education background.
So the Chinese carrier program, taken from a point of view of both military effectiveness and national prestige, is a sign of China's technical ability on the whole. While technical and scientific ability are developing rapidly and are reaching the levels of what is found in the world's most industrialized and developed economies, there are still large technical and logistic problems to be solved before China can have an effective aircraft carrier program.
A naval intelligence paper written on the matter of China's naval program.
And another one. Notice the rather sober tone of both the navy's official take on the matter.
Another official military paper, and again one that seems to not treat the matter as much of a threat.
http://www.washingtontimes.com/news/2011/aug/1/china-begins-to-build-its-own-aircraft-carrier/?page=all : a newspaper article on the launching of the Varyag.
http://www.cbc.ca/news/world/story/2011/08/10/china-aircraft-carrier.html : another newspaper article. Both of the newspaper articles seemed to be slightly more alarmist in tone than the naval reports.
http://www.snopes.com/photos/military/chinacarrier.asp : Snopes' report on a forwarded e-Mail about the Chinese supercarrier, which is obviously very alarmist.